I live just south of Lake Union (2200 Westlake building). There are several condos around that have some rentals available. They'll be a much higher quality building than an apartment of similar price.
If you want more of a social thing, then an apartment may be better. There's a building that's pretty popular called Rollin Street Flats that has a good scene.
Stuff's expensive around here, especially with Amazon moving in. Unfortunately you're not going to find a great place for under $2k.
Belltown is also pretty close and has good night life. Easy bicycling distance. There's a relatively decent complex called 2900 on First that should be in your price range.
Craigslist is a good place to look if you decide to go the condo rental route.
Good luck, and welcome to Seattle!