As CAsoldier puts it, training can be concurrent. If she already knows the basics of using a handgun and just needs to learn how to use one defensively, I'd move up to the Personal Protection Inside the Home course, otherwise start with Basic Pistol.
For martial arts, I hesitate to specify a particular art, I find it's more important to find an instructor who focuses on self-defense and mindset. Even with BJJ, you get schools that are focused primarily on competition, not street survival. A BIG handicap of MMA/BJJ is the emphasis on the ground game and submissions, if you're in a fight for your life, the last thing you want is to go to the ground, especially as a female. You need to know how to get out of holds and grapples, but mobility is essential. Females generally are completely outclassed, even with martial arts training, when it comes to up close and personal with adult males (doesn't help that many times other male students go easier on female students, either because they're reluctant to hit a female or because they're attracted to them, or they're just poor training partners and go along once the female starts a move). I like Hapkido or old-school Judo or Jiujitsu for joint manipulation and chokes. I'm fortunate in that the school I train in does a hybrid of Tae Kwan Do and Hapkido, so we cover both distance fighting and close up, the main thing we're lacking is ground fighting. So I consider it more important to find a good instructor who focuses on practical self-defense than one who tries to emphasize that their art is the be-all and end-all of martial arts. Avoid belt mills and schools that emphasize competitions, they both teach bad habits that can get you hurt in a fight.