For most bike adjustments, a 4/5/6 mm allen key and screwdrivers (phillips & flat head) will do for modern bikes.

You can buy some nice tool kits from Park and others, but you may find you get may tools you never need or use, so it can be better to buy as needed some good additional basic tools are

Chain breaker
Cassette lock ring tool / chain whip
Bottom Bracket tools (for the type you have)
spoke wrench for the size you have
Torx wrench if using disc brakes
Pedal spanner if you change pedals often
high quality cable cutter

headset spanners / cone spanners are also useful if you bike has an older headset / cup and cone hubs