You definitely should go down Alpine when you're there! If I weren't already going somewhere, I'd meet you there.
If you want to day hike in the Tetons, try Cascade Canyon via Hidden Falls. You can take the Jenny Lake ferry to cut out 2 miles, and it's a nice trip across the lake.
String Lake to Leigh Lake is a nice for SUPs that you can rent in Jackson. Or take the Tram up the Jackson Hole Ski resort and hike down (or tram it down again). Views are spectacular.
On the Idaho side, you can hike up Table Mtn (all day deal) or any of the other day hikes. Mountain biking is big on that side, too. Ask at Fitzgerald's bike shop in Victor, Idaho.
Eat at Bankhok Thai or Teton Thai (if you're on the Idaho side: in Driggs, Id) for dinner and Shades in Jackson for breakfast or lunch. Try Japanese in Jackson at Nikai. Great sushi.
Where to stay depends on your budget. There is a youth hostel in Teton Village...or you can stay at Amangani (ha, ha).