Hey Bretagus,
Welcome to the board, I second Gary's comment about finishing school first...
It is going to depend on whether you decide to join as an officer or enlisted, there has been some restructuring this past year and you may be more limited in your selection of assignment (type of units) as an officer.
By all means go and speak to a recruiter about your options and if you do not get a warm fuzzy feeling from the recruiter find another. Make sure you feel like your working someone with both your and the Navy's best interest in mind.
As enlisted you will have control over the type of diver you become to some extent:
EOD - explosive ordinance disposal divers, you will not get as much bottom time normally as some of the other diciplines and you will be diving re-breathers and SCUBA. There are many school available to this field (fun stuff airborne, SEER, small arms, ect.)
SEAL - rough road, long training and apprenticeship (about 1 year to become qualified) again you will have the ability to attend a ton of different types of courses and be diving re-breather and SCUBA
Diver - diver is again a rate in the Navy (for a period of time it was a skill badge, like an additional duty with another source rate you competed for promotion in) lots of different rols within this group. Salvage units, ships husbandry units, submarine tenders, ect. Amount of dive time will vary greatly with assignment. In later portion of your career as a diver you will be able to try and get assignment with a saturation unit (NEDU or Deep Submergence)
Sea-Bee - the primary focus for See-Bee is construction, you will get training in demolitions and various construction trades.
You could also look into Army, Coast Guard, Marine Corp (combat diver only), or Air Force (PJ only) and see who will make the best offer to you.
Make sure you get your choice included in your contract and make the best of you time in the field.
PS - Gary is a former Navy Diver, I am a former US Army diver but spent a good deal of time working with Sea-Bees and instructing at the Navy dive school.
Good Luck - Deep Sea!!
Jeff