Lifelong resident of the Kenai peninsula here.
Black bear, black bear everywhere. you can road hunt, atv or hike in along the road system fairly easily or you can take a boat across kachemak bay from homer, Limit 1 a year to non residents there, 3 a year in the rest of the unit. if you decide to go to the south side of Kachemak bay make sure your water taxi has a transporters license. but id recommend hunting here over the road system overall especially in fall or spring. less people and in my opinion better odds for a 6 foot plus bear. if you go in the fall lots of trails get you above alpine where all the good berry bears are.
there is moose hunting but access to the best areas is kind of difficult, ,most draw tags are for residents only, the harvest tag (OTC) is both residents and non residents, but comes with antler restrictions.
caribou is a draw tag
and all other big game requires a guide.
Spruce Grouse and ptarmigan can be found pretty much anywhere. hope this helps