The range in rentals will be a reflection of size of apartment or home, how close to transportation, how close in to Seattle you need to be, do you have pets, just to name a few. If you are comfortable living in West Seattle you can get a studio for about $450-$550, one bedrooms for about $800-$900, 2 bedrooms for $900-$1400 and up it goes. Prices in downtown Seattle are considerably higher and can range from $1400 to as high as you can imagine. Capitol Hill has many older apartment buildings and prices start at about $900 for a 1 bedroom.
This is just a sampling of the variation. As others have stated Craigs List and SeattleRentals.com are great places to start your search. If you are looking for someone to help guide you through your search you might want to find a property manager, one of whom I can heartily recommend by the name of Steve Glover and you will find his contact information by clicking on the link below.
Best of luck to you!
Otherwise, towns north or south of the city (Bothell, Lynnwood, Shoreline, Renton, Burien) will also provide affordable options.
In respects to utilities, I believe most landlords will cover water/sewer/garbage but tenants are usually responsible for electricity, gas, cable and phone.
If you are coming to find a job, I'd suggest you try to find a place close to where there are lots of job opportunities in your field. If you know where you will be working, then try to find a place as close as possible to that.
You should be able to find a share situation for $400 to $500 a month.