Which of these fish, vegetables, minerals, monster loot, and artifacts are useless and can be safely sold? Which ones are only used in bundles (which makes them useless to me, since I signed up with Joja Mart)?
There are only a few cases in the Community Center bundles where you need more than one of something:
Construction Bundle (all 4)
Quality Crops Bundle (all 4)
5 gold-star Parsnips
5 gold-star Melons
5 gold-star Pumpkins
5 gold-star Corn
Adventurer's Bundle (any 2)
10 Bat Wings
1 Solar Essence
1 Void Essence
Fodder Bundle (all 3)
Other than those, every other bundle only requires one of each thing.
So with the above exceptions, you can guarantee you'll be able to complete any bundle by just holding onto one copy of anything you happen to produce (crop, artisanal good, caught fish, etc.).
Not every single unique thing will be needed in a bundle, of course; but since food doesn't rot or spoil, you can always sell your excess later after the bundles are completed.
Of course, you might want to hold onto two of each thing, so you have one for Community Center bundles and one for any quest that might come up.
Unlike most recent Harvest Moon games there isn't a list of various important items you would never know you should keep unless you played the game before. The game doesn't suddenly require that you have 100 pieces of some otherwise useless junk in order to unlock some key feature. If you've already gone into the Community Center and seen the stuff you need to collect for the bundles, you'll see the stuff you need to keep, and in most cases you can dump it off there as soon as you get it.
About the only exception is if you're late in opening the Community Center and turning in a couple of the easy bundles, you might not realize you need certain things you can only (easily) get in Spring. In that case you'll have wait one extra month until next Spring before you can finish off all the crafts room and fish tank bundles and get what they unlock. You'll probably have to wait until a least next Spring to finish all the pantry bundles because one its bundles requires quantities of crops that are hard to acquire in one season.
(While you do get rewards for turning in each bundles, only turning all bundles of a category unlocks anything. The bundle rewards are all things you can get in some other way.)
Otherwise the stuff you should keep is pretty obvious. Everything has a use, but some things are more important than others. You'll need to keep building supplies for constructing new buildings on your farm. You'll want to keep ores (or better yet refined metal bars) for upgrading your tools. You might want to keep some crops for when you upgrade your house and get kitchen. Or just sell them all, it's not that big of deal.
Monster parts are probably the only thing that it might not be obvious you should keep. They're useful in various different crafting recipes, and you can end up using a lot of certain ones. So hording them can save you multiple trips to the mines to get what you need. They don't sell for a lot, so there isn't much temptation to sell them anyways.
Sure every once in a while you'll get a request of the board for some crop that only grows in a previous season, but not being able to complete isn't something you should worry about. The gold part of reward will usually be for about as much you can sell it for, so you're just missing out on some relationship points. Points that might not be worth time it takes to go back home, get the item and then track down the person to give it to. Also certain requests require that you mine or fish for the items requested even you already have them.
I know I'm late to the party, but I found this site through searching and figured someone else might stumble across this too looking for answers.
If you plan on fishing, bug meat is important in crafting most baits. I sold off tons of it during my initial runs in the mine and didn't know until it was too late. So I had to start going back in to kill the bugs in the first 20 levels of the mine to grind for more bug meat to continue fishing with efficiency.
That's probably the one thing I didn't realize I needed to keep and sold it all off initially, thinking it was useless. You don't need bait to fish, but it reduces waiting time for bites and it's what you need to fill the crab pots, if you intend on using them as well.