Get a good air rifle. A gas piston rifle in .22 will take small game. A .25 or larger pneumatic rifle will get bigger stuff. Most of the pneumatics (except the large calibers) have barrel shrouds that make them pretty quiet.Chapman’s guide to airgun hunting is on the web as a pair of PDFs. Do a search, lots of info.
Definitely better than a slingshot. Crosman brought its old pneumatic pump .22 pistol back with a carbine stock - as part of something called the "Doomsday Backpacker Bugout Kit" with backpack, water bottle, first aid kit, etc. All good marketing fun, but that pistol has been dispatching rats and bagging rabbits since the late '50s.