Welcome to shooting and welcome to the forum. My advice is.... don't buy a gun yet.
Since you mentioned that there's an indoor range nearby, go over there and talk to the folks who run it. If they act like they don't have time for you, find another range. Hang around a little and watch. Pick up what you can from everybody there.
If you have a local gun store, the owner or folks working there can also be good sources of information. Take your mom with you and handle both a semi-auto and a revolver. If they don't have time for you, go find another dealer.
Many times they'll have good used trade-in guns that might save you some cash to start with. And with a used gun the shop might be willing to trade it back for another one, if you or your mom decide after shooting it that you'd prefer a different type gun instead.
If you have a relative or neighbor who's "into" guns, they can often be a good source of info, but be careful. Not everyone practices good gun safety, unfortunately. Way too many people get killed by supposedly unloaded guns. Use some discretion in who you rely upon for advice.
Especially since you don't plan to carry right away, don't make a decision yet between a revolver or a semi-auto.
Since your mom's going to use it for home defense, a revolver might be simpler and less intimidating for her to learn and own.
As for caliber, many folks use .22's for self defense, but most want something a little larger for self-defense. Either 9mm or .38 would be a good choice, but use "mild" target-type ammo to begin with. You don't want your mom to hate guns because of the recoil of hot loads. And stick with the ammo you train with to start out. Don't use target loads at the range; then load up +P "self-defense" ammo for the nightstand if you haven't trained with it - and especially if your mom's never shot hot loads. Target loads will "do the job" quite nicely at the distances of most self-defense shootings.
Make sure she's "into" this plan as much as you are. Involve her from the beginning, because you don't want her to be afraid of guns or afraid to use one if she has to. You should buy a gun that she's comfortable with, not one that you simply like.
Try to find the money for you both to take some sort of basic training course. They'll provide the weapons. What ever gun you choose, you both need to learn good, basic firearms safety procedures first.
As you go along you'll soak up information like a sponge, and that's a big part of the fun.
Good shooting!