A large set of screwdrivers, and pliers, is an essential start.
For these (and spanners, but not adjustable wrenches) look for tools made of Chrome Vanadium steel, which is a hard alloy. "Chromed" steel or Carbon steel is a cheap substitute that will not last. It will be stamped on each tool, and emblazoned on the packaging. It's useful to have Stubby drivers as well as regular length.
You can usually get a comprehensive set from places like Screwfix and Wickes. You will need flat slot screwdrivers, and Pozidriv in three sizes. You are unlikely to need Phillips unless you have an old car or move to America. Beware of sets made for the US market, like Stanley, which often have Phillips drivers instead of Pozi.
The little "bits" that go in an adaptor or cordless drill can be useful, but you need real drivers as well.
this is a comprehensive set
don't get this one, it has Phillips not Pozi
good value
If you ever wear a driver out (this can happen if you use the wrong size) buy a really good replacement to fill the gap in your set. You will find that you have a couple of sizes you use al the time, and others you hardly ever use.
pincers for woodwork
good set of pliers with plastic grips, if you need them
for undoing plumbing things
inexpensive toolbox If you break it, get a more expensive one next time.
claw hammer
You will also need an electric drill. An 18v Combi with two batteries will do most jobs. If you can't afford that yet, a mains one is cheaper and more powerful and OK indoors where you have a socket.