Important Bicycle Maintenance Tools
This isn't an exhaustive list of tools but they should get you by most maintenance and repair jobs.
Get yourself some cleaning liquid and soft cloths. You can buy cleaning products specifically for bikes or as I often use, just general purpose cleaning liquid, like washing up liquid. It's still fine for de-greasing and washing off all the crud and grime. Make sure you check the liquid is safe to use on rubber, alloy and plastic though as you obviously want to avoid damaging the bike.
You probably have a set of screwdrivers anyway but check you have a selection of both flat head and crosshead (also known as Phillips) screwdrivers. I personally like to use a battery powered electric screwdriver but these are obviously an extra expense that many people may not want to use.
Have a short and long pliers. Try to also have a very tiny set of pliers.
You'll also need a wire cutter for various jobs. Many pliers also have a wire cutter element to them, so if your pliers do then you will not need a separate set of wire cutters.
You'll need a small hammer for various jobs where you need to hit something into its correct place.
Plastic Tire Levers
I used to use a flat head screwdriver to peel off bike tires but not anymore. The risk of damaging the inner tube is too great. Buy some plastic tire levers for this job. They make the job of taking the tire off a breeze and greatly reduce the risk of damaging the tube and rim.
Certain jobs such as calibrating the disc brakes need the help of a torch. You can either use a traditional hand held variety or you can try one of the torches that are attached to a headband, which allows you to continue working. You will be very happy you bought a headband type torch if your bicycle breaks down in the dark.
Set of Wrenches
Buy a full selection of wrenches as you'll need them for a wide range of jobs. You can also buy an adjustable wrench but I've only had partial success using these. More often than not, it's usually better to have separate wrenches of different sizes.
Brush and Toothbrush
A brush is essential for cleaning various parts of the bicycle such as the chain and a toothbrush is perfect for those hard to reach places such as gears. I have a word of warning though. Always lubricate the chain after cleaning it with proper bike chain lubricant from a bicycle store and not some household lubricant such as WD40.