How to chose via diameter and drill size based on trace width
The goal is to create a via with at least as much conductive area within the hole as the trace connecting to it (generally speaking, of course). My personal rule is to make the drill size diameter the same as the width of the trace, and the pad size roughly twice the diameter. This gives you a little bit of leeway in case your board is too dense to allow these sizes, and you need to adjust them. This is just a general rule that can be useful to beginners. It gives you a good size to shoot for.

Here is what completed vias look like on the board:



It is important to note that small vias will cost you quite a bit more than regular size ones. Generally I don't recommend going below an 8 mil drill. A microvia is a via that is less than 6 mil in diameter, and will cost you quite a bit more. 

Physical size (beyond the 6 mil "microvia" limit) really isn't that important unless you need to consider current-carrying capability or controlled-impedance. Once these come into play there are a lot of things you'll need to consider such as plating type, plating thickness, plating length (thickness of your board), via positioning, etc. In basic designs, however, where you just need to bring one trace to another layer, I would suggest using 8 mil for all traces smaller than 8 mil, and for thicker traces use the trace width for the drill diameter. It's just a good rule of thumb.