We just moved here from out of state without ever visiting here. We went from hotel to hotel until we found an apartment so we had the opportunity to see ALL parts of Seattle including the suburbs. I had a little more at stake so had to be a little more particular. Since you aren't as concerned about crime and want to be in the city, I'll tell you the best places to look are on craigslist (make sure you can google the sites and look around the area) and also apartmentguide.com I found mine that way. Just because they say they cost a certain amount means nothing. We have a $900 apartment we are renting for $750 in the Renton area. So, there are a lot of specials right now.
If you would be willing to go as far as Everett, there are GREAT apartment deals there right now. Also, Lynnwood is great. Loved that whole area.
Shoreline is sketchy. There are good areas and bad areas. The rent is reasonable there. I will tell you what everyone else will tell you, stay away from White Center. Be careful on transit and don't take the tunnel transit. You are a midwesterner like I am (Indianapolis). It is a LOT different here. People are very cold. ICE cold. You won't find the openness here like you do in Ohio (I have family in Dayton, been there quite a few times). It is just different here. We are staying in this area until our lease is up in one year and then moving to the Olympia area. It is more of a small town feel with more of an artsy crowd. You may decide you want to do the same after you move here. It doesn't rain all of the time like they say. BUT, it is cloudy most of the time. So, you have to take your multivitamin that has D in it.
If I seem like I am being a Mom here, it is because I am in my 40s and am a Mom. LOL! I have a good friend from high school who moved here from Indiana. He has lived here for 14 years and has never gotten used to the culture here. You can feel free to ask me questions as you have them.