Hello, Rick!
You are going to love staying at Disney's Animal Kingdom Lodge! (You may never want to leave!)
To get to Mickey's Backyard Barbeque Dinner Show at Disney's Fort Wilderness Resort & Campground, I'd suggest that you drive, since you have the convenience of your own car. While you certainly can get from your own resort to the other resort via Disney transportation, this would require at least one transfer each way, since most resorts do not have transportation with links directly to other resorts (there are, of course, some exceptions to this, such as those resorts on the monorail line or which are connected via water taxi.)
As you pull into Disney's Fort Wilderness Resort, which is considered a Magic Kingdom Resort, simply tell the Cast Member at the gate that you have dinner reservations, and you'll be allowed in to park your car.
This road map of Walt Disney World will help you to see what an easy drive it is from Disney's Animal Kingdom Lodge to Disney's Fort Wilderness Resort. Just look to the left side of the map to locate both resorts.
Please send any additional questions you may have as you think of them; I'd love to help! Enjoy the BBQI)