For hunting, you'll have to go in the fall when hunting seasons are open. Also, be advised that non-resident hunting licenses are really expensive. In some states (Wyoming is one), non-resident big game hunters have to be accompanied by a licensed guide.

Non-resident fishing licenses can be expensive, too. However, many states have 7-day or 10-day licenses.

Since you're South Dakota residents, why not start with the Black Hills?

I'm moving this to the Trip Plans, Reports and Inquiries section, which may get you better answers. Browsing through this section might help you, too.