Well, he might conceivably have picked up some of his pistolcraft during his stint with Villa and Co. in Mexico. He did get to see some good pistoleros (e.g., the young George Patton) in action.
It's been a while since I saw it, but IIRC Indy was using some fairly sophisticated and little-known (even today) one-handed shooting techniques in Raiders. For instance, keeping the firing-hand elbow slightly bent, but with the flexor and extensor muscles (biceps and triceps) slightly tensed against each other, to create a "shock-absorber" effect, both for recoil control in rapid fire, and to kind of "float" the weapon a little bit when firing while moving, especially laterally or diagonally.
Indy also seemed to generally fire from eye level whenever there was physically room to do it. I can't say for sure he was aligning the sights in the conventional manner, or just looking through them, or just glancing over the top of the gun, but he seemed to be getting some sort of visual index (at least a quick, coarse one), which is almost always better than any sort of unsighted "hip-shooting."
As for his hand-to-hand fighting skills, it's harder to say. He was a big, strong guy, naturally athletic, and I'm sure that helped. I'm not sure, but it seems like I heard somewhere that he learned to use a whip from some circus performer.
He used sword-like weapons effectively in Temple. Whether he had ever studied any sort of edged weapon combat (such as sabre fencing, or the related art of singlestick), or was just doing what came naturally, I couldn't say.
There was a good deal of close-up, hand-to-hand fighting with every sort of edged and blunt weapon in the trenches of WW I. There were a number of experts in French savate around then too. He could have picked up a few tips from them.
This is a matter which merits further study.
Merv