For a small order, I'd suggest working with a company that does both board fabrication (drilling, plating, and etching) and assembly.
There are lots of companies who offer that kind of service. Personally, I get boards from Advanced Circuits (http://4pcb.com/
) and they do have an assembly service. By all means check around and see what other companies are out there though. When you find one that sounds good to you, talk to the salespeople, let them know you're new to the process, and find out what they need from you in terms of information, parts, and files.
Fair warning: hardware has a significant startup cost. In most cases, having ten boards made will cost about as much as having a hundred made. In both cases the bulk of the price is due to setting things up, you're just distributing that cost across different numbers of boards.