What's in the toolbox?

It’s a disposable world, right? Nobody fixes things; everyone just tosses them out because nothing is fixable anymore.
Not true. Witness the popularity of repair cafés. There are still items out in the world that can be fixed, and there are houses that can be fixed, and you might be the one considering fixing them.
Why?
Maybe you can’t find someone else to do the task, or maybe money’s a worry. Maybe you want to try before tossing something in the trash.
Your first step is to make sure you have the tools you need before you get started in a burst of enthusiasm and have to pause partway through for a dash to the local hardware store.
The internet is your friend here. YouTube has tutorials on almost any project you can imagine. (You have to assess your abilities, of course. For example, doing your own electrical work is usually a bad idea.) Reddit has subreddits on every topic imaginable, including home repair.
The old-home basic tool list
Got a leaky faucet or a stain on the wall? Repairs can be taken care of in minutes if you have the right tools.
What those tools should be varies depending on the handyman. Consider this a starting point, and acquire more tools as your skills advance.
Where to look: For many tools, yard sales are absolutely a place to try; you can get quality tools for very little cash. Your local hardware store will carry much of what you need.
Remember to be sensible and careful. Respect your tools. Pay attention while you work. Whatever cuts or puts holes in something is also capable of cutting or putting holes in you.

The tools

Stepladder. For dealing with projects overhead safely.
Tape measure. They come in a variety of lengths, but the standard 25-foot is a good start.
Hammer. For hammering nails into a surface. A claw hammer has a flat head for hammering and a claw for pulling out nails.
Nails of varying sizes. Nails are good when you need some flexibility if force is applied to the surface. They’re also useful for finishing work because once in, they’re flush with the surface.

Drill. Power drills are lovely. Alternatively, you can take the old-fashioned approach and use a brace and bit or a hand drill. Augers are good for holes in the ground or in ice.
Make sure you use the correct bit for your project. It all depends on what you’re drilling into.

Screwdrivers. Phillips (cross) head as well as a large and a small slotted screwdriver. For turning screws.
Screws. They clamp onto surfaces better than nails, but they don’t have much flexibility. Usually you have to drill a hole first.
Nuts and bolts. Very strong fasteners. Often used in appliances and other metal things.

Spirit level. Shows whether a surface is vertical or horizontal.
Wire cutters. Self explanatory.

Saw. Saws come in many flavors, from hacksaws powered by you, to table saws powered by electricity. There are chop saws, chain saws, jig saws and more. For cutting wood or metal, depending on the saw. They can also cut you, so be careful.
Sandpaper, coarse to fine. Use for removing finishes or smoothing away splinters, or getting a wall ready for painting or plastering.
Paintbrushes and other painting equipment. For painting. Obviously.
Toilet plunger. Hopefully you have one. Useful for unsticking a unflushing toilet or reducing the water level so you can clean the toilet.
Plumber’s snake. Breaks up clogs in drains.
Duct tape. Wikipedia says this was originally made of cotton duck fabric and was called duck tape. Now made of other substances. Strong, flexible and very sticky. Tape a lawnmower seat, tape a package, tape all kinds of things. Actual ducts use a special duct tape.

Wrenches. Use on pipes (a pipe wrench) or nuts—for which you’d want an adjustable wrench or a socket wrench. Use the right size and type for the job.
Rags. For cleaning up after you’re done for the day. Your local hardware store can suggest chemicals (or alternatives) you might need to help with that cleanup.