Here's Dave R's primer. If you were an analytical-type guy, you could plot all calibers on a 3-axis grid.
On the "front" axis, you have "big & slow" at one end of the spectrum, and at the other end you have "small and fast". The third axis is either energy or case volume (how much powder will it hold). That axis also related to level of recoil, and price.
Generally speaking, what you want to shoot at, and how far away you want to shoot it, will give you a short list of calibers to pick from.
You want to shoot small things up close? .22lr. Want to shoot small things far away? .223, .22-250 and things of that ilk. (Shooting paper counts as a small thing).
Want to shoot big things up close? .45-70 and things of that ilk.
Want to shoot bigger things far away? .416 Rigby, .338 Lapua, .50 BMG. ($$$$)
Sitting right in the middle of that 3-D matrix, and best overall compromise, is .308 or .30-06 (there will be lots of disagreement on this.)