We live in the Beacon Hill neighborhood, which is just south of downtown. There isn't much going on in our neighborhood, so I wouldn't recommend staying there. Neighborhoods that are close to us that we would recommend staying in are Downtown, Belltown, Capitol Hill (towards the east side 15th-20th) and Lower Queen Anne. Green Lake, Ballard, Fremont and Eastlake are all nice neighborhoods as well, but they are just a little farther from us.
The airport is about 20 minutes south of downtown. It's about a $25 cab ride from the airport to downtown, plus tips and fuel surcharges. Be aware that the ceremony is not downtown, so you will need either a rental car or a ~$25 cab ride (from downtown) to get there.