You'll get lots of advice on here. For your first game day experience, you can't go wrong with The Strip. I take a group each year to Tuscaloosa, many of whom have never been, and we tend to hang in that area.
Let's say the game is 2:30. I would get to the Strip around 10:30, hit up The Houndstooth. Great memorabilia there and a fun-loving atmosphere. You can have a couple of drinks and watch the 11:00 games on ESPN. They also have fantastic Wolf Brothers BBQ you can buy right there at The Houndstooth. We love the BBQ nachos. After that, you have a couple of choices. I would recommend heading toward the stadium and checking out Gallette's, a large gathering place that has famous Yellowhammer drinks. Being a non-SEC game, it shouldn't be ridiculously crowded. Try a Yellowhammer and hang with fans. Then, shortly before 2:00, head into the stadium and watch the lead-up to kickoff.
After the game, we like to hang at Egan's. It's a smallish bar with lots of folks. Has a real dive feel to it. That's my advice. I'm sure others have great places too. RTR.