All horses do not need shoes. Depending on what you are doing with the horse and if he has any existing soundness issues he may or may not need shoes. Many older horses with ringbone need shoes and a good corrective farrier to keep the feet properly balanced, and the horse sound. Many competition horses' feet go under much more strain than a normal horse, and they need shoes. Etc.. but for your run-of-the-mill pleasure horse, they should not need shoes.
As far as the first aid kit, I have boarded at "full-service" barns before, ie you can leave the country and your horse wont even notice you're gone, and you still need to have your own kit!! At first, you can use your trainer's stuff, but as you get more experience you realize that you need a lot more. We have a vet kit that can deal with most ailments at the barn and it fills up 2 cubic feet of tupper ware. Some basic things that are good to have:
Access to warm water, or a kettle
A foot soaking bucket
Syringes (huge is the best size, and a few medium ones)
Lanolin or other emolient based dressing
Spot on insect repellent (the dog stuff works great, check with your vet first though)
A Mixing cup and mixer (spoon, knife, chopstick)
Access to running water
And from your vet it is useful to get:
For some stuff you may need other things, your kit will grow! I may have left out a few things, but this is what I find myself using the most!
Also having good resources and a good vet helps. I like Equus magazine, and the horse dot com for resources and my vet is always available on the phone. The Science of Wound Ointments for Horses The Horse: Your Guide to Equine Health Care