Jools, you will indeed need to buy the correct sized tap, but you may need a couple of different ones, depending on the job in hand.
There are three basic types of tap. A taper, or 1st tap for starting the thread off. A second tap and a plug tap.
If you are tapping a through hole, and it's not too deep you could do the job with a taper tap on it's own. You may need a 2nd tap if the hole is deeper.
If you are tapping a blind hole, then you start with the taper and finish with a plug tap. The plug tap is cut with a full thread right from the nose of the tap.
You'll also need a tap wrench to turn the tap with.
The hole needs to be drilled according to the thread size. Yes it will be undersize, but the actual diameter difference depends upon the thread size.
A M4 thread for instance will be drilled 3.2mm, a M10 will be drilled about 9mm off the top of my head.
Lubrication is always a good idea
It'll stop the tap breaking the metal away rather than cutting it, so as to give a good sharp thread rather than a rough one, which ultimately will be weaker. Use a cutting or specific tapping lube.