The best way to finish challenges is to wait until you've finished the main story missions and you've cleared all the levels of all the activities. By this point in the game, you'll probably at or very close to level 50 respect, and have enough cash to afford at least the "damage immunity" upgrades and "infinite explosive ammo/grenades" upgrades. You'll also probably have cleared some of them just by completing the objectives in other game modes. Trying to max out most of these challenges before that point is just going to be a waste of time.
Be warned that there may be minor spoilers below, so if you haven't finished the story and the side activities you might want to stop reading here. Nothing major, but still, you've been warned.
ONE BIG FAT WARNING
STAG Commandos stop spawning after you clear the game. Some people have reported that you can get STAG Commando kills by replaying the final mission. I've also seen a co-op bug that if you don't finish the game on your own save (ie, you use the co-op skip feature on the final level) you won't be able to replay the final mission. If this happens, you'll have to co-op with someone who still has the ability to spawn Commandos, or be ready to load an earlier save.
Survival is key
By far one of the most time consuming ones is the Survival challenge. This is time consuming for 2 reasons: one is that it takes a while to complete each wave of each survival, and the other is that you can't decide when one starts. Always answer your cellphone when you're given the option, and make sure that you're going to the location and staying within the radius for the entire challenge. Don't bother bringing a chopper or VTOL to these, as it's somewhat glitchy about how high off the ground you can be and still stay within the radius.
If you've got most/all of the damage immunity and it's a timed survival, just hide someplace and perhaps mark a few of the Saints in the area as your homies. Let them do the work, and revive them as needed. You can hide inside buildings, behind low walls, underground, etc, so long as you stay within the red circle on your map.
If you're working on a "kill x number to get to the next wave" survival challenge, you'll want to focus on blowing up gang cars. Kill off or dismiss any homies, as their kills don't seem to count. If you've got a co-op partner, their kills do count.
Each survival is unique, in that you'll get a different call from a different person inviting you, and they'll send you to a particular location each time they call. Even if you answer the call, it's possible to "miss" the challenge by ignoring the blue map icon for long enough. If you miss the call or the challenge, they will call you back again later.
To get survival challenges to spawn, you'll have to be on foot and not involved in another activity. (see Is there any way to force a survival challenge to start?) I recommend doing the following while you wait:
Group 1: Hostages and Surfing
Hostage taking - When you get into a car that has passenger(s), you'll be given the option to press a button to start the hostage diversion. I found the best place for this is near the Rusty's Needle tattoo shop that is northeast of Saints HQ. I don't know why this intersection is so profitable, but it is. Run in a circle around the intersection and look for cars. Parking minivans and other 4-door cars in the intersection sometimes helps spawn more of these. You don't need to complete the hostage diversion in order to get credit for taking hostages. As soon as you start it, you can jump out of the car and you'll get credit. You get +1 to this counter for each person in the car when you start the diversion. You can't do hostage diversion with gang members or police, including your own gang. They'll just get out of the car. You can't jump through either side window without kicking any front seat passengers out of the car, so make sure that if you're going to jump into the car, you do so through the front windshield. Passengers in the car don't spawn until you're looking at the car and you're fairly close, and they despawn quickly after the car leaves your field of vision, so don't chase cars that you missed but appeared to have passengers; you'll just find the car with one person in it when you catch up, usually. This one's in the top five most painful, in my opinion.
Vehicle surfing - if you get to the "lucky intersection" and you don't see any cars with passengers, find a car with a flat hood and crouch on it. When the car starts moving, you'll be shown the proper button to start the car surfing diversion. If the driver doesn't seem interested in moving, a shot through the window at the passenger's seat usually gets them "motivated." You'll have to maintain your balance on the hood to keep the diversion going. 30 seconds to a minute is a good amount of time per attempt, but you'll need a total of 10 minutes to compete the challenge. As long as the car is moving, it doesn't really matter what the driver is doing. They can swerve left and right, speed up and slow down, etc, and the balance minigame is pretty much the same regardless. Probably the most boring 10 minutes of the game, unfortunately.
Group 2: Looks like we're going to have to JUUUUUMMMMMP!
Base jumping - Go to Saints HQ, and get to the penthouse via the front door or the garage elevator. Run past the pool and the helipad, and jump straight out down the street. Use your parachute, and then you'll be given the option to set a target. If you jumped straight out from the helipad, the target's always in the same place, under some elevated train tracks. You can either land on the tracks, or go underneath them to hit the street level, either counts. For the most part, you can just pull back on the stick to slow your forward progress. You shouldn't need to move left or right. Jack a car and drive back to Saints HQ to start again.
Car torpedo - When you're approaching Saints HQ to reset from the base jump, hold the "exit car" button while going at high speed to start the car torpedo diversion. If you've got other cars parked in front of the HQ from previous attempts, you'll easily get high scores.
Windshield cannon - If you've maxed car torpedo when resetting for base jumps, slam the car into the front of the building at full throttle instead of jumping out. If you've already done the base jumps, start the Insurance Fraud diversion and drive the wrong way down the highway. Hitting other cars is very likely to fling you out. If you can manage to get thrown completely off an elevated highway section, you'll go a long way towards finishing this. When you almost run out of time in the Insurance Fraud activity, pause and exit the activity. For more info and a video, see: How can I most efficiently launch myself through my car's windshield?
Group 3: Stupid Car Tricks
Big air time - at a certain point in the plot, the bridges between the islands will go up. Get a fast car, give it max torque and the nitrous upgrade, and then go full throttle at the raised bridges. With a fast sports car, you can probably hit 250 per jump.
Wheelie - to do a wheelie, get a motorcycle and accelerate while pulling back on the "move" stick. Be careful, though: you have to land cleanly to get credit for the wheelie toward the challenge, just like you do to get respect for it. (This means no crashing until you touch your front wheel down. You can also mess up your wheelie by tipping too far back and having your back fender touch the ground, but it doesn't always cost you your entire wheelie.) The easiest bike to do this with is the Sandstorm or the Estrada, although by the end of the game/activities you'll have the "Cyber Blazing" bike in your garage permanently. Some bikes are too heavy to lift, so if you're on one that you can't tip back, ditch it and get another.
Stoppie - to do a stoppie, accelerate a bit and then brake while pushing forward on the "move" stick. You'll go up on the front wheel and slowly come to a stop. This is pretty easy compared to the wheelie.
Stores Robbed - Can I rob my own stores for challenge completion?
Streaking time - While streaking, can I "shock" police officers?
Two-wheel Distance - What's an easy way to complete the "2-wheels" distance challenge?
Gang members taunted - See Can I get credit for lewd gestures at people who are already trying to kill me? - if you can't get gang notoriety anymore (ie, you've got 100% of the properties owned, gang ops destroyed, etc), see the "(Gang) Specialists Killed" section below. This is a super painful one as well.
Nut shots - What's the most effective way to make Nut Shots?
During all of these you'll likely spend time wandering around on foot. If you still don't have all of them, just stand around in a quiet area and wait for the phone calls.
Post Survival Cleanup
These challenges you likely got organically during the story missions and the survival challenge farming, but I'll do a quick overview of them and how to farm them should you find yourself lacking:
- Distance travelled - Fly around in the VTOL or any other airplane if you've not travelled far enough yet.
- People run over - Drive on the sidewalk. Bring a tank if you want to farm some other challenges at the same time.
- Oncoming lane distance - Drive down the middle of the highway going the wrong way.
- Near misses - Drive down the middle of the highway going the wrong way carefully. :)
- Powerslide time - Press the powerslide button while you turn. If your car is fast enough, you might be able to drive in a circle with the powerslide button and the throttle held down to do this continuously.
- (Gang) Specialists killed - The only hard part about this is getting gang notoriety. You can replay side missions to get some temporary notoriety and to get some gang members to show up. Trafficking and mayhem missions in the regions where the gang used to control work best. Once you have notoriety, cancel the mission, but make sure some gang members are still around that you can kill to regain their ire.
- Vehicle Mod Spending - Take a bunch of cars to a Rim Jobs or your crib garages, and blow money on them.
- Human shields used - Grab everyone you can find and hold them with the human shield button. You can throw them or snap their necks immediately after grabbing them.
- Multi-kills - Destroying gang or police vehicles with a tank, rocket launcher, or other heavy weapon will give you a multi-kill.
- Grenade kills - Throw grenades. Infinite grenades helps.
- (Cars/Helicopters/Boats/Tanks/Emus) Destroyed - Obviously you can destroy enemy vehicles, but you can also use vehicles from your garage. Just take them out of the garage and destroy them, and then repeat. For helicopters, crashing the copter works fine. Note that VTOLs aren't counted towards this, so high STAG wanted levels won't help.
- Mascots killed - Prof. Genki's side mission is loaded with mascots, and one of the survival challenges is mascots in Emus. Two (sometimes literal) birds with one stone.
- Carjackings - This only counts cars that you stole from people, so cars parked in lots or on the side of the road don't count. Stand in a busy intersection and relieve people of their cars.
- Hoods explored - Drive around. This is probably one of the first challenges you will unlock.
- Shortcuts found - Watch for pathways between buildings that connect major streets, and drive through them. You only have to find about a quarter of the total shortcuts in the game, so it's likely you'll get this easily.
- No weapons kills - Equip your fists and beat down gang members or police. This is much easier once you're mostly invulnerable. Note that even if you've got all the damage immunity levels, you're still vulnerable to melee attacks, so be careful in close quarters lest you end up in the hospital.
- Stag commandos killed - In the story mission "Air Steelport" there are tons of these guys to kill. If you still need more, max out your police notoriety. They're relatively rare, but they do show up from time to time. If STAG is no longer in your city, you'll have to replay the final mission to encounter more STAG forces.
- Compliments paid - Just complement random people on the street.
- Gang ops defeated - You have to get near a gang operation for it to show up on the map. If you're missing a few, once you own everything and have all the activities completed, check parts of the map that aren't tinted purple. A quick drive around the area ought to turn the operation up.
- Kill brutes - Get your gang noteriety up (see "specialists" above) and brutes will spawn. You'll also see them in Prof. Genki's side mission.
- Headshots - Aim high! Use the pistols or an assault rifle for the best accuracy.