First: Deep breath. Panicking never helps.
I would leave the plants in there, it's good for Mom to have plenty of cover, and if you plan on raising the babies in tank with mom, it'll offer them protection.
If you plan to remove the babies, set up a small tank with an airstone (I've used a 2-3 liter soda bottle to start off), place all of the babies in there, and don't bother feeding for at least 2 days while they absorb their yolk sac. Once they're ready to eat, just grind up the food in your fingertips until it's a fine powder. and they can eat whatever mom is eating.
Siphon carefully with an airline every other day, and once they're big enough not to fit in anyone's mouth, they can return to the main tank.