There's a full weapon list here, and most weapons support upgrades, which you can find a list of here. I'd suggest focusing on one weapon early on and getting it to level 4. By the end of the game, you will have enough cash to own each weapon in each category. At one point you'll also be given the choice to take a weapon upgrade cost discount, so if the upgrades seem overpriced, keep playing the story missions.
Also note that in the upgrade menu on your phone are various upgrades for enhancing your bullet carry capacity and reload speed, and these are quite useful.
You can save a good amount of money by picking up enemy weapons. Even if you love the one you've got, picking up and dropping one an enemy left behind will unlock that weapon in your crib and Friendly Fire for further use/upgrades.
Some personal observations/strategy:
- The Penetrator is available early on, and it's surprisingly good at dealing damage to both people and vehicles in the game.
- Late in the game you'll have the option to take the Apoco-fists, which give you the ability to explode people with a punch, and slam into cars like a brute.
- Either of the early pistols is worth it, I chose the Kobra but others have suggested that the Shepard's explosive bullet upgrade makes it better against brutes.
- Pistols are super accurate, even at long range, which more than makes up for their small clips. Headshots and nut shots are easier with pistols for this reason.
- I picked the D4th Blossom as my primary SMG. It does decent damage at close to medium range, and can be good when you've got a big group to deal with. It's not terribly accurate though, so I considered it to be at least a 3rd or 4th tier weapon.
- Late in the game you can unlock a "Decker" SMG that I believe does good damage, but I haven't played with it much yet.
- It's been noted that the Tek-Z10 SMG sports an incendiary ammo upgrade. If you're having trouble against soft targets or vehicles, you might consider it.
- My general strategy with the SMG class was to sweep left to right at head level and hope some of the bullets counted as headshots.
- The Ultimax is the one I picked in this category. It fires fast, reloads fast, and does high damage at short range.
- The Grave Digger is not, in my opinion, as good a weapon. It fires more slowly and just isn't as good in a pinch. It's been noted that it has an incendiary upgrade, like one of the SMGs though.
- There's an "energy" variant of the shotgun you'll encounter late in the game, which tends to overheat too quickly for my liking. I used it for a bit before going back to the Ultimax.
- The Krukov is full auto, and I didn't much care for it.
- I used the AR-55 (burst fire) for a long time until a better rifle came along.
- The Viper Laser Rifle is an energy weapon, so it has no reload but if you fire too long it will overheat. It fires fast and does high damage though. It's the winner as far as I'm concerned.
- I only had access to the Annihilator, and I found it to be good, but keep some money aside to get it to level 4 early - that gives it lock on capability which is essential against flying targets.
- Late in the game you'll get a remote detonated charge, if you make a story mission choice to take it. Since the Annihilator is probably the best anti-air/anti-tank weapon, I can't say that it's worth taking them. If they were an upgrade to frag grenades though (like they were in Saints Row 2)....
- Flashbangs at level 4 are required for the "Stay Classy, Steelport" achievement. They're quite useful when running up against gang operations, as they stun a wide area of enemies immediately upon impact. They also work well against Decker specialists, who tend to warp around the area otherwise.
- Standard grenades are super useful for blowing up groups of enemies and taking out vehicles.
- Molotovs don't do enough damage in a range to really be useful unless you're out of frag grenades.
- I didn't really find a use for the electronic grenades; I asked my co-op buddy if he used them at all and he said "only when I pushed the wrong button and equipped them by accident"
There's a wide variety here, and you'll unlock them slowly as you play story missions. Personally I preferred the RC Possessor, simply because at level 4 it can take out any military vehicle with one shot (and explode it to take out other targets). Plus it's one of the easiest ways to get some of the flying military vehicles in the game. If you shoot a VTOL/chopper with it, and land the craft, the pilot will generally jump out and leave the keys in the ignition.
The sniper rifle is a required weapon at times (Guardian Angel missions, primarily) and it's pretty good as snipers go, no sway and 3 zoom levels. It's not really a snap-shot weapon though, and it isn't as useful in as many situations as the Annihilator is. You're more likely to have at least a half dozen guys on you, or a fast-moving brute than you are likely to see one guy at range that the pistols can't handle.
Some of the other special weapons will be most useful in the areas where they're given to you (the airstrike and missile drones come to mind, for instance). They're fun to play with, but from a practical perspective aren't that useful.
When you start to get up close to level 50, infinite ammo upgrades will be available for all weapons. Grenades are excellent for this, especially combined with explosive resistance. Second on the list is Annihilator ammo. Once you have explosion immunity and infinite explosives, you are an unstoppable killing machine.
Once I had those upgrades, I went for infinite pistol ammo, because it's a high damage, high accuracy weapon that's got a wide variety of uses. Compared to the explosive damage of the Annihilator though, it's a distant second or third.
There's also an achievement for having a level 4 upgrade to one weapon in each class. Note that I believe this includes grenades, and you have to upgrade the baseball bat (the only melee weapon that supports upgrades) in order to get this.