Re: Help Please
If you're looking for that time of year you're limited a bit unless you hunt critters that aren't regulated by season.
Spring bear and spring turkey would both be open, and require you to have a tag for each species as well as a hunting license. Depending on the area you hunt you may have to put in for a draw (lottery) to gain a tag also.
Depending on where you go for bear you could also potentially hunt without a tag (but most likely still need a hunting license):
Exotic deer, antelope or similar species
As for what is required for a non-US resident, not sure. I'd figure out where and when you're going bear hunting first and then contact that states Fish & Wildlife service for more details.
Now if you go hunting for bear in the fall there would be more options for other game animals to hunt (of course dependant on where you hunt):
Deer: Whitetail and Muledeer
A variety of game birds, upland and waterfowl
And some harder to obtain tags: mountain goat and bighorn sheep
As well, the exotics, but those wouldn't really be "fair chase" animals. You'd have to hunt them on a private ranch.
I'm sure I've missed some and that the other guys will fill in. Keep us posted on your adventure.