A lot depends upon in which branch he served during what years.
The 1973 fire at the National Personnel Records Center destroyed the majority of US Army and USAAF/Air Force records from the early 20th Century to the mid-1960s. A small number of US Navy, USMC, and Coast Guard records were also destroyed, but nowhere near the same amount. The records which were destroyed had no known copies made, so if they were destroyed, they're gone.
That said, the National Archives are a mandatory stop to see what you can find. The link Canoerebel posted is how you go about starting the formal request. If you can luck out and get a hold of any portion of his service record, then of course that will be the best source of reliable data. Be aware that they only give information with the veteran's written permission if he/she is living and only to a next of kin if the veteran is deceased, so you may need to involve others depending on your situation. The Archives folks will be able to provide specific guidance.
There's one other thing to try for official records if his were lost in the fire. It can be somewhat of a 'chicken and the egg' scenario, but if you know that he served overseas at a US base, especially in Europe, there were independent records kept there. The process for exploring that route depends on the particulars of each case. In addition, if you can ID at least one unit to which the veteran was attached at some point, the vast majority of units have well-established histories that you can at least glean general info from (where the unit was stationed & when at the very least) which then can shine more light on other pieces of info you have specific to your relative.
A very helpful thing to do once you've made a request for official records is to gather up any documentation, pictures, medals, awards, etc. that you can. Copies/photos are OK too, you don't need originals necessarily, just the clues/information they can provide. The best bet for photographs is early in the vet's service, typically during some phase of training they received. Cadets/trainees almost always get their picture taken at some point. If the most 'obvious' documentation (for example, discharge papers, otherwise known as a DD-214) is not forthcoming, then think of any associated documentation which might exist. Think of if the veteran was married when they served, and if the spouse lived on base(s) with the veteran at any particular time. Spouses are also commonly issued ID cards, PX cards (for the on-base store), and whatnot as part of living on base. It's also another person with a separate personality - sometimes a vet who may not have been the best at keeping records has a spouse who was much better at such things (opposites attract, right!?).
Then start searches online using whatever bits of information you were able to gather. If there's anyone you trust with personal info, you can have them look as well for a 2nd pair of eyes. If you need help identifying things which aren't too personal, perhaps a unit patch, a ribbon/pin on a uniform, you can give selective information on a forum like this one so folks who are familiar with the military/history can give feedback.
I've helped others with the process, let me know if there's anything I can do to help.