There are generally three routes I’d suggest.

A) Compact 9mm

I’d personally suggest a Walther PDP, but something like a Glock 19 or S&W M&P9 Compact would be fine handguns.

The advantage to this option is that generally the triggers are easy for a new shooter to learn quickly, and the size allows for good control of the weapon and a good capacity. They are also very customizable in general. Models for optics are available, and equipping them with a weapon light is easy.

The disadvantage is that they are a little bit larger and will require extra consideration to carry comfortably, and if you don’t carry comfortably you likely won’t carry consistently.

You can have one for about $500 which will allow money for a holster and extra ammo. I would highly consider an optic, but it is not necessary. An optic can flatten your learning curve though, showing you issues with grip, trigger pull, and recoil anticipation that you might bot see with irons, on top of being a superior sighting solution and easier to see in general.

B) Sub-compact 9mm

As much as I hate suggesting Sig products, a P365 might be one of the better options for this class, but I’d personally go for a S&W M&P Shield or Walther PPS M2 here.

The advantages here over a compact would be solely their ease of carry. They are a good bit smaller and lighter. Those same advantages though manifest in their disadvantage of being harder to shoot accurately, as the size and weight exacerbates poor form/technique. Put in enough time for discipline and training though, and you can overcome the issues. With discipline and training though, you can shoot the larger compacts that much better.

3) Revolver

Don’t consider this option lightly.

I’d suggest a S&W 640 or SP101 in general given your budget. Either of these options will eat up most of your budget after a good holster and a few boxes of ammo. You could get a S&W 442/642 for around $500 and have plenty for a holster and extra ammo, but they are harder to learn on than the aforementioned models.

The advantages here is that you have surety of fire, a simpler manual of arms, and potentially a great training aid.

The disadvantage is relatively less comfortable recoil, lower capacity, and more technical reloads that are slower.

I train kids in competitive pistol shooting, and I generally start them out on a revolver if I have them directly under my instruction and they haven’t provided their own handgun. You can dry fire like a mad man with them, and training the double action only trigger pull can clean up grip and anticipation issues that are very common in new shooters.

The past two years I have carried revolvers almost exclusively when I am not on duty. My autos are starting to collect dust.