Allow me to start with there is no "best" (or "worst") slingshot with the exception of poor build quality that is unsafe to use. Lots of shooters are amazing shots with "cheap" frames or tree fork naturals that cost them nothing to make other then time. What's "best" is a matter of opinion to each individual shooter. This is because everyone's hold hand/style is slightly different. What fits the hand best, is typically what shoots the best since its much easier to get the same hold consistently each time.
For band and ammo combos for hunting I can be a little more specific. Still there are several options that boil down to shooter preference alone. This is still very much a matter of individual preference sport. Please make certain that you are a good enough shot before attempting hunting.
Bands:
Some main things to keep in mind are shot placement is key. When slingshot hunting know that the energy will be lost over distance. What may be a capable bandset for out to 15m may no longer be adequate at say 25m, even though you can still hit what you're aiming at. The outdoor air temps can dramatically change a bandsets performance. What may be good during the warm day may be too weak on a cold night. For my hunting bands I like to stick with latex in the .66-.8mm thickness depending on the time of year (the thicker used in the coldest temps). As far as tapers I like to use a 12-20 or a 15-20. The rest of them I make at in between 20-15 and up to a 25-18 (for my hunting setups not for target). Just remember the latex and taper being used will depend on the ammo size/weight being used (as well as other factors like the temperature). In colder temps a thicker latex may perform best and yet be totally awful when its warmer outside. All my hunting bandsets I shoot at a 570% elongation if it's A 600% max. If not seeing the power you want and you can pull past your anchor point then cut some off. This can increase shot velocity dramatically. Difference in draw lengths will change a bandsets performance as well. Also, a longer draw can utilize a narrower taper.
Ammo:
It all works as long as it has enough energy and is put in the correct spot. A ton of preference here as well. I'll try to be as brief but informative as possible.
First, I'll cover lead ammo. Lead has the best transfer of energy due to its heavy weight but remember it's toxic, harder to find these days, in some places illegal to hunt with, and way more expensive than steel. If not making your own then you can buy lead muzzleloader/black powder ammo round balls. The best sizes are .32 caliber up to .44. A .32 caliber round ball is 8.4mm and a .36 is 9.5mm and thus a very common one. The .44 comes in at 11mm.
Second, is steel ammo. Steel is lighter than lead but still very effective due to the higher velocity that can be achieved. Some really accomplished shooters choose to use 8mm steel ammo for hunting applications. Its even lighter weight allows for some very high velocities to be achieved. However, at further distances the light weight does not have a high enough amount of energy for hunting. For that reason 8mm IS NOT RECOMMENDED for hunting! The 8mm is very light and until ultra precision accuracy is achieved should NOT be considered as hunting ammo. The 9.5mm-12mm steel are the most common. If you have a chronograph, testing the speed and putting velocity into a ballistic calculator app will tell you how exactly how much energy your bands and ammo combo are producing at any given distance. A great tool for determining what your maximum effective hunting distance is or should be.
In summing up, I'd like to say that I personally start with a .66mm thickness or higher, with approximately a 20-15 taper and 9.5mm steel ammo as my starting point. This then may need to be adjusted from there according to suit personal needs or wants. As a general rule, you want to have enough power to be able to put anywhere from a rip to a hole in a steel soup can. Remember those are the U.S. style can and that can strength does vary greatly.