Hi Angel Eyes.
Keep it simple to start with. Your budget sounds good, and you can find some great bikes. There is not a great deal of difference between manufacturers at this level. Almoast all of these bikes have threaded eyelets to attach a luggage rack and child carrier.
You dont need any special bike clothing, but whilst you are learing the ABC, its best to wear long sleeves and leggings, to avoid grazing your skin. Cycling gloves are really useful in protecting the palm of your hands from the tarmac, and a helmet may prevent an injury turning nasty.
As a complete beginner, rather than a born again cyclist, you may need a few months to become skillful enough to carry a child. I think the trailer idea may be safer. Handling a bike at slow speed is the most difficult part of cycling, and a heavy child makes the bike handle a bit odd.
Make sure the bike fits you well. Many women try to ride bikes that are to big, and that is no fun. If you get a mountain bike, get the shop to swap the tyres for some with a smooth/slick tread. They are so much easier on the road.
You sound like you have plenty of local help, so good luck, and let us know how you do.