What RJay said. These are called the "H&R Self Loading Pistol". H&R used that name because they had already described some of their revolvers as "Automatic", because they ejected empty shells automatically when opened.
Yours is in 32 caliber; 32 ACP, also known as 32 Automatic or 7.65mm Browning. There was also a smaller 25 caliber model. Ammunition for your gun is not too hard to find, although you might have to go to a sporting goods store instead of discount store like Walmart.
These guns were well made of good materials and are considered safe to shoot if in good condition. I have one of these guns, and in my opinion, it shoots surprisingly well, better than you would expect, since it looks somewhat awkward. But the grip fits my hand well, at least, and the sights are OK for the period.
Loading an automatic pistol's magazine can be tricky if you've never done it before; you need to sort of slide the cartridges in from the front instead of pushing them straight down.
Dry firing the gun (pulling the trigger when the gun is unloaded) should do no harm in moderation, and would help you get familiar with how it works. (It is better to have it uncocked before you take it apart, I think.)
This gun has three safeties: the manual safety lever, a grip safety (the spring-loaded hump on the grip, which must be pressed in to fire), and a magazine safety, so it cannot be fired if the magazine is not in the gun. Unfortunately, all these safeties do the same thing: they disconnect the trigger from the firing mechanism, rather than locking up the firing mechanism in any positive way. This is the only significant weakness of the design, IMO.
The only other weak point is the grips. They are made of hard rubber, which was a very common material at the time. It is OK stuff, but it gets brittle with age, and all of these guns are 75+ years old now. The part of the grip that goes under the manual safety lever is the easiest to break, so be careful if you take the grips off.
If you do a Google search for taking apart an H&R self loader or something similar, you will come up with a number of websites that will tell you how, including a YouTube video:
Welcome to the Firearms Forum, Dan S., and thanks for putting up pictures!