The USA is a big place, might help if you can narrow down which section you would possibly visit.
As to seasons, we have much narrow seasons than you do, so timing is everything. Bear would not require a lottery tag in most areas, and can be hunted in spring or fall (or both) depending on the state/province. They can be hunted from stands, stalking, bait, hounds, or as Friwi said, over a dump in some areas. The style of hunting varies with the state.
Mountain goat and sheep are almost certainly unavailable, since most require a lottery for the tag, and many people have been putting in for the lottery 10-20 years.
Be glad to try to answer any other questions you might have.
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