Coyotes are fair game year around.
Elk and deer can be hunted every year, and usually antelope tags can be drawn every year. With elk and deer, if you don't draw a specific area, you can buy a general license and hunt general areas. Bear and mountain lion are over the counter, but areas are closed when quotas are met. Turkeys are mostly on general tags, but addtional tags for specific areas are also available, and seldom sold out.
Moose, sheep, and mountain goat are tough draws, as is buffalo.
If you are in Casper, bowfishing is good at Pathfinder and I used to shoot lots of carp on the Sweetwater arm of the reservoir.
Lots of bird hunting for grouse. If you are a waterfowl guy, the Platte has good duck and goose hunting.
Hope you like the wind; I lived in Casper for 30 years, and it gets a bit breezy at times.