This prevents your foot getting stuck in the stirrup if your horse spooks. Having a smooth sole lets your foot slip out easily and quickly.It also prevents being dragged.
Boots are the best because they also protect the top of your foot. They're much tougher than runners. Getting steeped on by a horse can really hurt.
Also if you have some stretchy comfy pants, I found tackies or yoga/gym pants the best.
If you start enjoying the lessons I would think about buying a pair of half chaps/or full length ones (it comes down to personal opinion) and a helmet. Hopefully your ridding school provides helmets if you don't own one.
Nowadays when I ride I use sticky bum jodies, tall boots or half chaps with jodie boots, gloves and a helmet..
With all that said about safety I hope you enjoy the lessons!