You should be able to do everything yourself.
First off, go to any grocery store or home center and ask for a Mr. Clean "Magic Eraser". Buy 3 or 4 packages of those. Some places sell generic Magic Erasers a little cheaper, so if you can find the generic ones, they work just as well.
Dampen the Magic Eraser with water and scrub the dirtiest part of the floor with it, and see how much the improvement is. If you want it cleaner, or want to clean it faster, put on some rubber gloves and dampen the Magic Eraser with a mild acid like CLR. The CLR will dissolve the dirt and clay particles that are in the porous surface of the tile and grout.
After cleaning, you'll probably want to seal the floor to prevent it from getting dirty like that again. Look in your yellow pages under "Janitorial Equipment & Supplies" and phone around to find out who sells "S. C. Johnson Wax" or "Johnson Professional" products. Ask the company selling those products who the local S. C. Johnson Wax sales rep in your area is and how to contact him. Ask your local S. C. Johnson Wax sales rep what he'd recommend using over a terrazzo tile floor. You can also go to Johnson Wax's web site to find a customer service 1-800 phone number too.