Allow me to break it down a little further. By small game I assume you mean squirrels, rabbits, something you're wanting to eat. In that case go with a .22LR.
I own both and both are fun to shoot. No recoil, great for beginners.
17 HMR is a great round for groundhogs, varmints, and longer range shooting(out to 300 yards). It's more expensive to shoot than the .22, but also inexpensive compared to most centerfire rifle calibers. It's too much gun for squirrels, rabbits, etc... if you have any intention of eating them.
.22's of course are great. Everyone should own one IMO. Ammo is the cheapest around. Greater selection of guns available. Will not overly damage small game animals. The only downside is that your range in generally limited to 50-75yards.