I agree with ratherrapid that your filly needs more time, probably quite a bit more, before going to the training center. You'd be wasting your money to send her now and not doing the filly any favors either--after 2 months in a stall she is way out of shape. Much better, if possible, to turn her out on good grass in a group of her peers and give her six months or so to eat and build muscle and return to the peak of health. The added stress of going into training now is only going to set her back farther.
Just as a comparison, here's a picture of a filly we bred and kept to race. Picture was taken last July when she was a yearling, probably about the same age as your filly.
She didn't have the disadvantage your filly did--ours was never sick and was raised mostly outside, but you can see how much more muscle and conditioning she has. This is more what you're aiming for. Even so, we opted to give our filly more time to grow up and didn't send her to a training center until January of this year.
As for shoes, we never put them on babies unless there's a problem. The filly pictured above was still barefoot when she went to the training center and remained that way until a couple of months ago when she started doing some early speed work.