There are a lot of different routes to take.
If you sign up with nothing but a High School Diploma as a "Personal Health Care Specialist" (there is the new PC term for Medic) then you will go into training to complete an EMT-B certification. From there you go on to learn a lot of stuff that most civilian EMT-B's aren't allowed to do, I.V.'s, chest decompressions, ect.
If you do go in with a High School diploma I strongly suggest holding out until you can get a M6 ASI (Advanced Skill Identifier) in your contract, which will make you an LVN/LPN.
I don't know exactly what it is you're looking to do when you say Combat Medic - so I'll ask you, what is it you want to get out of the military itself? What do you expect to be doing as a Combat Medic?
If you're an adrenaline junky, and you think you're tough enough for it - you can get a OPT 40 contract which guarantee's you a shot at Ranger school. Although I hear OPT 40's are very very very hard to get these days. At the very least get an OPT 4 to guarantee airborne.
There is also the 18X contract, but to be honest with you it's just a way to get guys into Infantry. If you're going to be Special Forces - you'll get chosen from it throughout training, it's not something you can just accomplish because you got it in your contract.
From an Officer point of view, if you plan on going ROTC. You would want to get a BSN, and then you'd most likely be placed in the Nursing Corps.
Just keep in mind, unless you carry an actual MD in your title - everyone in the medical field gets deployed.
The only medics in general who actually see a lot of combat are Ranger Medics and Green Beanies.As an Officer in the sand box, your job is more of a management type position than it is leg work.
Most medics are encouraged to stay within base - usually you're forced to because there is way too much work to do anyway. Sooner or later just about everyone has to go clearing houses, though.
On a side note, the Navy Corpsman receive a lot more practical training than Army Medics do - it may be something you want to look into.