Start with a handgun. You can conceal and carry it and also use it for home defense. You cannot conceal a shotgun.
Do not go too small
The weight lives in the slide; the frame is light. Go too small and you get less to hold, less recoil absorption, and a shorter sight radius, which makes the gun harder to shoot and hurts your confidence at distance. For a first gun, stay compact to full-size, not sub-compact.
Fit, then lifestyle
If you cannot hold the gun correctly, nothing else matters, so fit it to your hand first. Then factor in how you will use it, carry versus home. Even a full-size gun conceals with the right holster and clothing.
Try before you buy
Take a class or rent several. Shoot a compact and a full-size, get your shots diagnosed, and pick what fits and shoots best for you. Do not rush it.
Budget and ammo
Budget about $600 before tax for a solid first handgun: Glock, CZ, Shadow Systems, Canik, Sig P365, or FN 509. The real cost is ammo and classes. Buy a box of about 25 hollow-point defensive rounds and confirm they run reliably, then a couple more boxes, plus at least 500 rounds of target ammo to keep on hand.
Storage and training
With kids, use a quick-access locked case (like a Vaultek): fast for you, not for a toddler, and teach your kids gun safety. Train ideally once a week with focused drills, and no less than every two to three weeks or you are starting over each time.
Get the carry permit, even in a permitless-carry state, for the legal education and because it looks far more favorable if you ever have to defend yourself in court. Buy, train, then take the permit class.