To get into any branch you have to have a bachelors in nursing. Not an associate degree, a full bachelors degree. You might check on USAF, Army and Navy ROTC programs because you might be able to get your bachelors paid for by doing ROTC. '
Once you get your bachelors degree you have to pass the national licensing test and have a license in at least one state.
If you can not do ROTC and have to pay for college yourself, you can get a direct commission into any of the three branches as a nurse as a 2 LT or Ensign. By going into the service, you can get student loans forgiven. You would have to check on exactly how they do that.
After you are in the service you will want to get into ICU nursing if you can. That will help get you into anesthesia programs. Assuming you do an outstanding job and apply for the service anesthesia school, you might get accepted and go to the military program, all paid for by the service and you get your regular service pay as you do it. When you graduate you will have to pass the national exam and then will owe the service 2-4 years of payback time. If you can not get into a program you can get out of the service and use your GI Bill and go to a civilian program. All the CRNA programs are master level programs so as far as "better" it is kind of academic. (Pun intended.)
The services can have as many company grade officers as they want so you usually make Captain/Lieutenant during your first four year tour. There is a limit on field grade officers so your promotion to Major/Lt Commander will be around your 10-11 year point. Unless you walk on water and are something really special, Lt Col/Commander is about all you can expect with a 20 year career. If you go regular you can stay past 20 years and can move up higher in rank. For a while the US ARMY had a CRNA as the Surgeon General of the Army. Can not go much higher than that!!
Forget going in a sub, specially if you are a nurse anesthetist. Room for women is limited and they do not need a lot of medical staff. They go out from their home base and often go under and do not surface until they return to their home base. They may be out to sea (under it) for months at a time. You would more likely be on a carrier, marine assault ship or hospital ship. Most of the time you would be sitting at a navy hospital so your assignments are limited. Keep in mind the more specialized you are in nursing, the fewer possible assignments you have available to you. Where most USAF bases have a clinic and clinic nurses, there are few hospitals where they have nurse anesthetists. If you want to do a lot of travel then get that out of your system early in your career.
About the best you can expect is assignment to Europe and you do all the travel on your own time with your own money. Same with Japan.
If you marry another service member then your life gets even more complicated as you both have different deployment schedules, different schools, different hours and there is no guarantee at all that you will even be in the same state, let alone at the same base. You will have to decide between career and family, taking assignments that will help your career VS taking assignments that are not so good so you can stay with the spouse. As an officer you have to not fraternize with enlisted as that is a career killer. Your life will be a lot easier if you got a civilian dependent husband. He can watch the kids.
Now, if you want to talk about anesthesia, the best military oriented anesthesia training is Army. They deploy close to the front and take the worst wounded cases. The Navy and Air Force are more or less back up so do not get as intense trauma training. The USAF does not deploy as much and tends to stay away from the front lines. If you want safer then the USAF is the one for you.
As far as the feminine side, they have strict guidelines as to makeup, hair style and color, nail polish and the like. You will have to comply with the requirements while in uniform. Forget getting any piercings except a single hole in each ear, forget tattoos. Besides the weight requirements you have to pass the fitness requirements. Tougher in the Army and Navy than the USAF but still, you will have to exercise at least twice a week to stay in shape.