I think that all depends on what you're talking about. As others have mentioned, you'll want some basic tools, like screwdrivers, brass punches, a torque driver, that sort of thing. Depending on what kind of gun we're talking about, mostly older ones, you'll need all that just for routine cleanings. A nice laser bore sight is also handy.
The next level of do it yourself gunsmithing would be something like mounting a scope. Very easily doable for the average person with the right tools. You would need an alignment kit, lapping kit, scope levels, and a torque driver. Installing sights is on the same level. Again, very easy with the correct tools.
After that would be bedding a rifle stock. Still very easily doable at home, but definitely more that could go wrong. This would probably be in the same category as installing a recoil pad, lengthening or shortening pull, adding checkering, etc.
After that, we're talking about doing trigger jobs and making small replacement parts by hand. That's probably the kind of thing only a real gunsmith should do, and only a good one. While it's certainly possible for the average person to learn it with readily available and inexpensive tools, the consequences of doing it wrong are pretty great.
After that would be machine work. A lot of what gunsmiths do is thread barrels, square off bolt surfaces, change chamberings, etc. But no, I don't think the average gun owner needs a lathe, unless you want to.
Nowadays, there's not much call for gunsmithing, what with the advent of CNC machines. If you know enough about firearms to troubleshoot problems, then all you have to do is find the malfunctioning part and order a replacement from the factory. Used to be, all replacement parts had to be hand fitted. Now you just drop them into place. For example, a Glock 22 might get sluggish on the ejection around 10,000 rounds. All you have to do is order a 15 dollar ejector from lone wolf and you're back in business.
If you wanted to learn a little about gunsmithing, I would suggest building a 1911 from a receiver. There won't be any machining involved, but all the parts will have to be hand fitted. That's about the closest you can get to real gunsmithing without buying 10k dollars worth of machines.