What should a first-time buyer get for small-game hunting?

A .22 caliber rifle producing at least 12 FPE at the muzzle is the standard answer.

A regulated .22 PCP gives the easiest path to consistent accuracy. A quality .22 break barrel works too if you commit to learning the artillery hold. Pair the gun with an airgun-rated scope, a good supply of mid-weight domed pellets, and a chronograph so you can confirm your real velocity and energy.