You won't go wrong with anything purchased from minitack.com
They sell a pleasure harness for $250 that is a perfect starter harness. It's the cheapest harness I'd recommend... Anything cheaper will be poor quality. If you find that you are really enjoying driving and want to really start putting miles in, you can upgrade to a more comfy, sturdy harness for your horses.
As for carts, the easy entry cart is usually where people start. They are cheap, sturdy, and do fit two. Again, as you drive more you may want to upgrade to a cart that is lighter and easier on your horses. The easy entry carts are about $600 new. Buy from a reputable dealer, and again if you spot them for cheaper somewhere it may be a brand that cuts corners and you may find the seat breaking in half or something...
Glad you're working with a trainer. If you buy the cart and harness they can surely get your little ones started.
Have fun,
Andrea