Hey dude, I'm 17, and I can relate to that loneliness thing.
One thing I will say, is try and convince your parents to let you start Martial arts. It worked for me, atleast. Not because of being bullied (although it will, in time, stop that), but mostly because the self confidence it builds is great, and you become very good friends with the people in your school of choice. Nothing is cooler than hearing some parent brag about how their child is spending money on clothes, and having your parents reply "Our child is learning how to beat up yours".
I used to be extremely depressed and lonely. I was overweight as well, and had just stopped talking to a friend of mine permanently (said he was my friend out of pity). Martial arts took care of the weight problem, as well as giving me something to look forward to and think about. Eventually I became very good friends with all the people I trained with; I was only 14, but was very good friends with a range of people between 12 and 32. Now I can't get enough, and rarely ever feel the pangs of loneliness or depression that I used to feel.
It doesn't have to be martial arts; anything that interests you is good. If you're really into computers, look into computer based clubs at your school. If you'd prefer outdoor activities, I still suggest martial arts, but theres no reason you can't try football or baseball instead.
The main thing is that a), you're in the presence of other people, b), when a person does something well they feel good about it, c) it's physically healthy, and d), both a and b are good for the human psyche. And don't say you're too shy to do it; I'm extremely shy, and my first martial art was Brazilian Jiu Jitsu. Watch some videos of BJJ people sparring; you're literally in contact with the other persons body with your body the entire time.