I'm a guy, but I'll give my thoughts:
If you can go to a shooting range, they sometimes have guns for rent. Shooting a handgun well depends a LOT on how well it fits your hand, and you can't know that without trying one out.
The best known Sig Sauer 380 is their single action 238. It is a good gun, but it is best carried 'cocked and locked', which is unnatural to a new shooter. I honestly wouldn't trust myself to carry one, and I've had guns for 35 years.
The Bersa 380 have an excellent reputation among the shooters I've met. It is a bit big for easy concealed carry, IMHO. The only way I've found to be comfortable with concealed carry is in my pocket, and my jeans don't have enough spare room in the pocket for the Bersa.
The Ruger LCP is a smaller, double action 380 that also has a very good reputation. It would be easy to conceal. Ruger builds good guns and back them up if anything goes wrong.
I carry a Smith & Wesson J-frame. Lots of guns stores recommend them to women, but I wouldn't recommend them to any first time shooter. They kick hard and cost $500 on up.
If you are new to shooting, you might consider a small 22. That is pretty small for self defense, but the only gun I've ever pulled for self defense was a 22 I had on me that day. Didn't need to shoot, either - the guys who had started surrounding me decided they didn't want to any more...
The advantages to a 22 are that a very small gun can be more than strong enough, and the ammo is cheap - about $3 for 50 rounds instead of $15-20 for 50 rounds. If you are new to guns, shooting a few thousands rounds is critical to knowing what you are doing. A small 22 revolver is easy to learn, and easy to shoot well. A Ruger LCR22
runs around $350-400, holds 8 shots, would be easy to conceal and lightweight (15 oz, and every oz counts when you carry).
My choice, which I would not recommend for most people (although I use long shirttails to conceal it):