There are tons of different ways you can approach this, and you have to figure out what best works for you. The obvious pathway for someone interested in the military and law is JAG, but that may not be the right path for you. There are a ton of reasons why it makes more sense to do military -> law school rather than vise-versa.
1) The GI bill. After three years of active duty, your law school + housing for three years is absolutely free at UVA, Berkeley, and UMichigan (of the T14 schools), and the rest of them do YRP, which means your education would almost certainly be free there as well. Plus, you'll still have an extra year left over. JD/MBA?
2) You get a not-insignificant boost in applying for law school.
3) You get a substantial boost in applying for jobs during and
after law school.
4) After 2-3 years in, you will probably qualify for funded legal education. Army calls this program FLEP, but the Air Force and Navy have similar programs. Not sure about Marines or CG. Under this program, the military pays for you to go to school (not using your GI Bill - you get to keep that), and in return you have to stick around and be a JAG officer for 6 years.
5) You may find after 1-4 years in the military that your interests have changed. This happens a lot to K-JDs that have really never worked full time.
6) You may find after 1-4 years in the military that you hate the military. You're probably better off figuring this out now than planning your whole law school career around JAG only to find you can't stand it.
That said, a couple reasons to jump straight into law school:
1) Direct commissioning programs are going to be easier than other entry programs. Army, Navy and AF's DC programs are like 5 weeks long.
2) Former officers report that their time in grade gets kind of screwed up because they end up going for boards a lot sooner than their peers, and subsequently with less experience. You won't have this problem.
3) If you're on a full scholarship from the law school, maybe you don't care that much about the GI Bill.
4) You start your legal career in 3-4 years rather than 6-8 years.
If I were you, I would skip enlisting and go straight for OCS.