See: View of Juneau from the top of the Mt. Roberts Tramway. A round-trip ticket on the tram will cost you $35. As an alternative, you can hike up from the downtown area (challenging 2.5 miles), enjoy a meal, the nature center and some native art at the top, and ride down for free (show the tram operator your meal receipt).
Do: State Capitol tour (you get your own free personal tour – very informative)
Eat (dinner): Salt (seafood in a modern, fine dining setting)
Eat (dinner, although they serve all meals): Rookery Café (small and large plates, fresh local fare, menu changes daily)
Drink (coffee): Heritage Coffee (local chain of coffee shops)
Drink (beer): Guided tasting at Alaskan Brewing Co. Since the brewery is a few miles from the downtown area, they provide a shuttle for $15 (pick it up at their downtown storefront). Tastings are free if you drive yourself to the brewery. We drank many Alaskan Brewing beers during our trip, our favorite being the Smoked Porter.