If you get 100 replies, you'll probably get 99 different answers.
My first handgun was a .22 revolver, then a .357 revolver. But, when I decided to get my carry license, neither of those were suited for concealed carry & I opted for a semi-auto. Basically more rounds & thinner pistol, so easier to conceal.
I lucked out & got one which needed no break-in because I didn't know enough about semi autos to even know they sometimes need break-in.
They can be ammo picky, sometimes need couple hundred rounds for parts to "mate". You get past all that, I believe a good semi-auto is the best for both fun at the range & self defense, especially if you intend carrying it.
As for brand, etc. I like Bersas & have them from .380 up thru .45ACP. But, I also am addicted to 1911s in .45ACP. You really need to shop around, handling & shooting as many different ones as you can, to decide what's best for you. I may own one in 9mm one of these days, after I get all of the .45s I have on my list.
As for caliber, many don't think so, but I believe the .380 is a viable self defense round with the right ammo. Those pistols however, tend to be blow back design & small, so susceptable to limp wristing, ammo picky & have a pretty snappy recoil. Due to size, weight & design, 9mms tend to have less recoil, except for the sub compact ultra lights, are more reliable out of the box & the ammo is cheaper. Then, the .40 tends to be pretty snappy also compared to the .45 so many go past it to the .45. Of course, those higher calibers mean larger & heavier so if carrying it is your ultimate goal, how you dress will enter into that.
Anyway, the research & the handling/shooting is part of the fun, so have a good time!