The amount of time needed for the connection is about an hour (for an experienced guy) to an hour and 1/2 (if the lines are copper or plastic). Since the labor rate varies and the extras vary across the world, its hard to give you an exact cost. Needless to say, in my area Merit shop contractors charge $99.00 per hr, Union Shop contractors charge $185.00 per hour.
Every shop will charge you a trip charge. Some will call it a diangostic fee, others a truck fee. It basicly is 1/2 hr travel (whatever the hourly rate is). Your fittings are the easiest part. Usually not more than $50.00. It will really depend though on any mark-ups the company will charge you. A $5.00 cost part, can sometimes cost $15.00.
Check out the local labor rates by calling a couple plumbing companies.