If you are new to working on bicycles you will be utterly amazed at how much you can do with just a 5 mm Allen key. Next in importance is a GOOD QUALITY cable cutter.

Don't cheap out when you buy those tools and don't expect them to last forever. As soon as they start to show signs of wear, throw them away before they start to round out your fasteners. A cable cutter that cuts all but one strand is a PITA.

When I was doing shop work I used to replace my Allen wrenches every year but I still have a cable cutter that's probably 50 years old and still cuts cleanly. Good quality tools might have you thinking twice the day that you pay for them but they will minimize your aggravation from there on out.

Bicycle specific tool needs evolve through time. I've got a collection of threaded headset wrenches that I'd gladly give to anyone who still needs them. Freewheel remover tools are vital but only if you have a bicycle with a freewheel that they'll fit. Crank and bottom bracket tools are another constantly revolving category. I'm still using my Park chain breaker from 20 years ago but, with ever narrowing chain widths, it's days are numbered. That $15.00 chain quick link plier was a smart purchase.