The issue you are going to have is the fact that this barn is 30 minutes away and you will have to go there 2+ times EVERY day. That will get really really tiring after a short while. It will be expensive too. My family did it once before and it was physically exhausting. Not to mention, you may have things come up that require you to be there more often than that. Also think about what would happen in bad weather, etc.
The care of a horse is quite expensive. I know I never would be able to afford a horse if my dad didn't pay for most of the care. Per month they cost several hundred dollars, and that doesn't include vet care and farrier expenses.
Someone mentioned leasing or half leasing. I think in your situation, since you would be responsible for all the expenses this would be your best option. I also suggest joining a horse 4-H club now (most clubs will take you even if you don't have a horse) and you will be able to learn and meet people who may be willing to let you use their horses. So many 4-Hers I knew while growing up often had an extra horse sitting around and they would let 4-Hers that didn't have horses use them. One year a friend of mine did not have a horse but wanted to do horse camp, so I shared mine with her and we took turns riding.
Free horses often need training, as someone else mentioned. I got an awesome free horse two years ago who is a doll, but she is green and still needs training. She didn't buck or take off or anything when we first got on her, but she didn't understand a lot of cues and she just needs time. In the hands of someone who has no experience training little things like this, a lot of damage could be done to both the rider and the horse. The last thing you want to end up with is a horse you are afraid to go near. One bad experience could ruin your love of horses for a lifetime.
Don't be discouraged though. Talk to your mom about some of the things suggested and go from there. I think it would be better than jumping outright into owning a horse.