On the surface this seems like a rather simple question, But imagine if the question was "What should I look for in hunting gun?" Essentially the same question, just a different weapon.
I found an article Now: Super Accurate Slingshot(Popular Mechanics Jan 1984) that talks about state of the art 30 years ago. Much like for guns there are entire web sites dedicated to the weapon The Slingshot Community as well as multitudes of articles. Google finds more than 7,000 hits for "slingshot hunting" in books. They are made out every material imaginable; wood, metal, composite. There are as many designs as there are firearms.
There are also several options for elastic/rubber bands used. Much like firearms, there are lots of options. The big online (non-hunting specific) have 100's of options.
When it comes to the weapon of choice, just like with firearms, personal preference and budget dominate the decision making process.
As for ammunition, it can be anything from found objects like rocks, to factory made round object like ball bearings, or even arrows.
In the end, all that really maters is practice. Pick whatever you want for the weapon, and the ammunition, just practice a lot with whatever you choose. You want your target to be dead on the first shot. Allowing an animal to escape with a broken limb is wrong. No matter what weapon you choose, how fancy it is or how much you spend, if you don't practice, you will not have consistent kill shots.