The glib answer is to look at what you know, then start looking for what you don't.
You're going to NEED to know the branch of the military your man was in. If it's the Army, you'll need to know the regiment and company. You can find these on Fold3, if you have a subscription, or Family Search if you don't (Family Search is free to use; Fold 3 and Ancestry are not, so that's usually where I start).
As long as you are on Family Search, you may as well see what else they have. The 1890 census of Civil War Veteran's and Widows is useful in finding out what happened to your guy after the war, and the census records are helpful in tracing his life before and after.
My next step WOULD be to send for his pension/compiled military service record. The former costs around $80; the latter, $30, except that the National Archives already had a couple months wait time BEFORE Corona happened, and it still not open yet. You'll be better off hiring someone to go look once they are back up and running.
Next, I would to the stupidly obvious step of googling the name and regiment. Then the name and the dates he lived. You can come up with all kinds of unexpected treasures that way.
Google to see if there is a regimental history, open it up online using Internet Archives and read about where the regiment was and what it was doing during the war.
Be sure to try alternate spellings if you come up blank. Records are only as clear as the penmanship of the person writing them and the deciphering ability of the person transcribing them.
And - another goofy idea - search for the name on Ebay. It's a long shot, but you never know (I found my great grandfather's glass negatives for sale there several years after my grandmother left them at the dump!). Leave a standing search for the name, so that if something does come up, they'll notify you.
Finally, ask for help on boards like this one. You might even find someone descended from you man posting here.
Hope this helps!