Definitely get proper riding boots, not wellies as your foot could get trapped, which is why the riding school got you to change to theirs - rubber will be fine though - Aigle and a number of others do knee highs - not too expensive, last forever and easy to clean! Also, if you change your mind on riding, they are still suitable for tramping over hill and dale, so won't go to wast.
Other tips:
Really don't be scared of falling - most of the time it's not a problem. Generally you'll find falling starts happening when you are pushing yourself to get to the next level. Rather than considering it as a 'failure' use it to analyse how to improve. One important rule though - let go of the horse on the way down! Horses prefer not to stand on humans, so if they are free to go, they try not to. If they are stuck by you holding onto the reins, they might have no choice.
Visualise making everything 'long and heavy on the seat-bones' when in a situation you're not sure of.
Don't look down, look ahead to where you're going.
Horses respond really well to balance and rhythm - so, if you want to go left, look left, then add your leg, ditto for right or straight. If your horse is trying to go a different direction than you want, ignore them and focus on / keep looking the way you want them to go - they will generally comply (eventually!)
Get in as much stirrupless/reinless riding as you can (including jumping) - it really pays off later.
Also, things really take time to come together - sometimes it will feel like you'll never get there, and then, five lessons later, the world is a beautiful place again.
After a lesson don't be surprised to find stiffness in places you never knew existed. 