I got my first two jobs through a staffing company, then was hired on by a staffing firm where I have worked for over 20 years.
Some agencies are great and others suck - you will probably want to sign on with a few, as another poster mentioned. Don't be deterred by the sucky ones.
Ours is a smaller, "boutique" agency, not one of the big nationals, so I think we tend to give more personal service. You might want to try a couple smaller firms, who frequently have a lot of small business clients, and a couple big nationals, who tend to work with the bigger companies.
Come in dressed professionally for your interview - that will be a huge plus in your favor. And don't be late - staffers and recruiters definitely notice these things.
I usually advise applicants to check in with their recruiter/staffer once a week when trying to get a position - not more or less. More and you are considered kind of pushy. Less and you might get lost in the shuffle. You might even want to ask your interviewer what would be a good day to call or email in.
Contrary to what another poster said, staffing companies actually operate on a very thin margin. It might seem like the spread between the bill rate and pay rate is huge, but we have massive payroll taxes (particularly unemployment), liability insurance, workers comp insurance, disability insurance, and background checking costs, among other things. On top of that, so many of our clients pay us very late - often 3 to 6 months out, and yet we have to pay all of our employees weekly. Sometimes I wonder how we stay in business.
Best of luck in your job search!