I agree with the observation, that if the main interest is rescue, then the Coast Guard may offer the best opportunities.
The first step is to visit an armed forces recruiting station. But do not sign anything on your first visit. Find the Coast Guard rep and the Navy rep. See what programs for high school grads they have to offer. Take home some literature from each, and make sure you do not sign anything. Tell them you "want to think it over."
Coast Guard and Navy programs like medical, swimming, diving, etc are very popular and so they get a lot of applicants. You need to be able to show them you offer something more than the average applicant (which is normally nothing). Athletics goes a long way. The medical programs like it when their applicants have formal first aid & CPR training, or experience of some kind, either volunteer (for the Red Cross) or part time (like working for an EMR company). The swimming and diving programs appreciate high school athletes, such as swimmers or track runners.
If the youth is an athlete, and has some experience and training, he/she stands a better chance of getting into the desired program. They like kids with motivation, who did something about it, to get ready for the Coast Guard or for the Navy.
When you get a written offer from the Coast Guard or from the Navy that specifically states there is a program you are signed up for, then it is time to sign up. Most new recruits, however, simply end up in the pool of general recruits, and do not get any guarantees of MOS (military occupational specialty) or billet (job assignment). A really good performance in basic training can get a recruit preference for a particular school, however, if the aptitude is there.
There is one further huge liability involved in the Navy's medical corps program. It has stateside duty, it has sea duty, and it has duty with Uncle Sam's Misguided Children (no offense intended however). Stateside duty at a major naval hospital is the best, of course. Sea duty involves several months at a time onboard ship, at sea, away from dry land for the first several years of a career. This is not for everybody. The worst in my opinion is duty with Uncle Sam's Misguided Children, however. This medic is a corpsman who provides all the medical needs of the amphibious infranty. It usually means you must hump and hump and hump, and then treat mostly heat casualties and heatstroke victims of The Children. The Coast Guard does not have to do anything like that.
You just need to keep all that in mind.
Good luck.
Anchors aweigh!