I have had many conversations about what a basic bicycle tool kit should contain. When attempting to answer that question, I have found that I answer differently, depending on the day, the phases of the moon, how much beer I've had, and how much people on social media are being buttheads and hurting my little feelings. So, this is my attempt to come up with a consistent and helpful list of which tools someone needs to do routine maintenance on their bike.

Before we go on, let me get some caveats out of the way.

This is not a list of tools you need to work on every bike that may poke its head up over the parapet. This is mostly a list of tools that I have found consistently useful for a decent variety of bicycles. If you're looking for the stuff you need to be a wandering minstrel mechanic, this is probably not the page for you.
There is no "correct" answer. Most everybody will have a different opinion, series of opinions, or the desire to tell someone else how thoroughly wrong they are because reasons.
I am not a professional bicycle mechanic. I haven't run a bike shop in central Bicycle Cycling City for twenty years or whatnot. I do fix up donated bikes for a local charity as time allows, so this list is based on my experience and my situation. Most of what I do leans toward the lower end of the spectrum. If you're looking for advice on maintenance of high end carbon fiber speed demon bikes that cost significantly more than my car, you may want to look elsewhere.
There is quite a bit of "grey area." The required tools are often bike- and situation-specific.

That said, I think this will get most people who are interested in maintaining their own bike moving in the right direction.

Step 1: Non-Bike Specific Tools

I often see people worrying that they will need to purchase a full set of top-end left-handed quill stem-adjustment spinney-cassette-wheelhub widgets for umpteen-jillion US dollars.

No. Not even a little bit.

I lay out a set of tools before every build. Tools that I know I will need on every single project. None of them are bike-specific.

A pick . You can buy them from any hardware store, but the one pictured here is actually an old bicycle spoke sharpened with a file and bent into shape. I don't know exactly when I'll need one, but when I do, it's best to have it to hand. And I need it on damn near every build.
Metric hex wrenches. Actually optional if your bike is old enough, but I will presume that your bike isn't an old Columbia 10-speed bought at JC Penny's (not that there's anything wrong with that...it's kind of awesome, actually).
#1 Phillips head screwdriver. Get a decent one here. I have no idea why, but the cheap ones suck.
10-inch adjustable wrench. Or the metric equivalent for all of you not in the States.
8-, 9-, 10-, and 15-mm box wrenches (again, unless you're working on a 1980's-vintage bicycle, in which case get the x/16- inch equivalent).
Cable cutters. Don't skimp on these. You want clean cuts when you're working with shifter/brake cable. Not-so-clean cuts don't work as well. Add to that, they're painful (hello, tetanus!).
Diagonal wire cutters.

None of these are tools that would be out of place in most home hobbyist tool boxes. No left-handed quill-stem-adjustment spinney-cassette-wheelhub widgets in sight.

Step 2: Okay, Now We've Arrived at Some Bike-Specific Tools...

All two of them. And one is optional.

Chain breaker.
Crank arm puller (optional if your bike has a one-piece crankset).

That's pretty much it. Granted, there are some tools you need based on your bike in the next two steps...

Step 3: Rear Wheel Gear Thingy Tools

This is that grey area I mentioned earlier, at least if you are not a professional bike mechanic (or holding yourself out to be one). If you are, you will need a bunch of various tools to remove the spiky things on the back wheel of the bike. None of them are particularly expensive by themselves, but having to buy a whole bunch at once can be a bit intimidating.

Don't blame me: that's the life you chose. On the plus side, if you're not a professional mackanick, then you only need one or two tools.

These tools will be specific to your bike. If your bike is somewhat WalMart-ish, or even low-end local bike shop-ish, you will probably need a Shimano-compatible freewheel tool. Other bikes, like fancy (old) Italian bicycles or totally-not-a-kid-thing BMX bikes may need different tools. Bikes with cassettes will need the appropriate tool, along with a chain whip. Why a chain whip you ask? Well, that chain needs to know who's boss, so there.

For more information about real wheel spiky things and the tools that love them, the evil capitalists at Park Tool have a very informative (and free!) page on that evil capitalist internet thingy here: https://www.parktool.com/blog/repair-help/determining-cassette-freewheel-type

Step 4: Bottom Brackets and Bearings, Oh My!

Another set of mostly bike-specific tools. On the plus side, changing out the bottom bracket cartridge or repacking the bearings should not need to be something you need to do on a regular basis. That makes this a very optional set of tools.

And frankly, if you have a one-piece American bottom bracket, you don't need anything more than an adjustable wrench and a flat-blade screwdriver, so you have a somewhat lower chance of needing a special tool.

At any rate, there are several bottom bracket types, so you should go here to figure out which (if any) tool you need: https://www.parktool.com/blog/repair-help/bottom-b...

Step 5: Well, That's It...

That is it. This is by no means an exhaustive list of useful tools (you can pry my torque wrench from my cold, dead fingers). It is an attempt to let people know which tools that they will need to maintain their own bikes. Certainly, if you are not confident in your skills, or you have a ridiculously expensive bike, you should take your ride to a local bike shop and let them do the work. But if you want to try to fix your own problems, then this will hopefully get you started.