Cool. Another Texas transplant! Me and My girlfriend just moved to Seattle from Houston about a month ago. So far we love it!
We chose Kirkland which is a city about 15 miles northeast of Seattle. It's next to Bellevue, Redmond and Juanita. It's not a bad area, although when our lease is up we may find something closer to Seattle, something that has a more neighborhood feel.
We haven't been here long enough for me to give you any great insights or suggestions. I would suggest Ballard, Fremont or Wallingford as i've heard good things about them. Also search on Craigslist. Also, the stadium district in Tacoma i've heard has some old duplexes etc you can rent.
We ended up choosing one of the larger apartments just because it was easier to find when searching from Houston. Rent in my area at least is not too outrageous. We live in a 2 bedroom apt and it's 1300 a month. If the same apt was in Houston, it would probably be around $1000 a month or so, (although my apt in Houston was only $625)....
Anyway, hope this helps. And welcome to Seattle/Tacoma when you get here.