Is it possible to serve in multiple branches of the military ?

I did. I first entered the Army in 1979 and served till 1986 working in MI (Berlin, Ft. Meade) and then entered the Air National Guard in 1986. Except for a break sometime around 1994/95 when I made a brief change to the Army Reserve, I wound up serving 14 years in the Air Guard. I went into the Air Guard specifically for the advanced training in satellite communications electronics I received.
In the Air Guard, I served in three squadrons, 2 of which were Combat Communications Squadrons and the third was an Air Control Squadron. I saw more of the field working in these tactical units than I did when I was in the Army. There are parts of both services that I miss, giving me a split personality at times ;-)
Serving in more than 2 services may give you a bigger adventure, but I'm not sure if you'll advance as far as you want to. But maybe that's not that important to you. The other problem is that some of the other services may require you to go through basic/boot camp again, which is not exactly what I'd be looking to do.

I entered the Marine Corps in 1980 did
four years after which I enlisted in the Navy and did eight years. I finshed in the Air Force retiring with a total of 21 years.

Not sure I could offer any good advice on how to accomplish this goal, however.... I do know of individuals that have served in three separate branches after 12 years of Active Federal Service. I believe they had breaks in service, which I don't think had much, if any effect on their transition from one branch to the next.

Yes you can serve in other branches of the military after you are discharged from your current service branch. You will have to check though because there are other factors at work here. I haven't seen any go from the Army to the Marines, but the other way around. My uncles started in the Marines and Army respectively then both switched to the Air Force. They both also served in the Korean War.

I started playing off with 2 1/2 yrs in the Army Reserve, I was able to get released and then spent 20 yrs in the Navy.

I did. I served 4 years of active duty in the Air Force as an aircraft hydraulics journeyman, and after I got out and moved back home to Texas, I joined the Army National Guard for 6 years. One thing I can say is there is definitely a difference between branches. You really see that when you serve in more than one. Me personally I enjoyed the Air Force more, mainly because of the job I had while I was in

Yes you can as long as you have a good discharge. My brother was drafted into the Army in 1970 and in the mid 70s he joined the Marine Corps Reserve.

I started my career in 1974 in the Army. Got out in 1981 and went into the Air NG. From there, I went to USCGR and in 1985, the Navy came to me. I went into the Navy until my retirement in 1998. I have a total of 24 years in the U.S. Armed Forces. Would not trade a minute of it. I should add that I stayed in Law Enforcement those 24 years and became a federal officer from 1999 until my retirement in 2011.

I think you need to focus on whichever branch you join first and foremost, and the reasons you chose that. It's not like working all the departments at Target, you should be building some love of whichever branch you join.

I joined the Marine Corps and loved it, but when I got out there were no reserve units close enough to me. I joined the National Guard because I had friends enlisted, and I ended up retiring from the Guard with almost 25 years total. I enjoyed the Guard, met some tremendous people, but my first love will ALWAYS be the Marine Corps.

Do yourself a favor, focus on whichever branch (I think you said "Army".) If it's the Army, be Army Strong, Be All You Can Be (Just don't be an Army of One, those folks suck!)

Just one at a time, I have known people that have changed service branches, that's their choice if they chose to do so. Myself I enlisted in the Air Force and never had any intention of changing to another Branch of the service and didn't. One friend however, was Navy, came to USAF and later returned to Navy, He felt more at home there. Worked with a few that went from the Marines to Air Force. Some with Army backgrounds went to other services or from other services went Army. Although each branch serves the United States once You have found a home and are comfortable with it there are adjustments to any change. Each person has to decide for themselves how well their skills will transfer into another service, how far back may this put Military career progression and what is to be lost or gained by the transition. You have to choose whatever path You are comfortable with but whatever We have in common there are differences. Keep in mind there are hundreds of different jobs You could have in each individual branch of the service and maybe cross training is better than finding a new home. In the long run a decision You have to male Yourself. Whatever You do don't do it on an impulse, look into it first, research then decide.