In the world of manners, accepting an offer of kindness and then imposing yourself on the person making the offer by demanding more is rude.
Your mother didn't ask you to take your sister, she demanded you do it. That was out of line.
If your bike was taken away because of this incident then it is too bad that I must pronounce your parents insane.
I agree with TonyCtattoo, never again make an offer to get anything for your parents again. If at some time in the future they ask why you never make such offers, remind them of the time they took advantage of you and then punished you for it.
If you can get a part time job now and begin saving your money, you will be able to move out on your eighteenth birthday. If you don't plan to move out at that time it would still be a good idea to have that money ready for the first big fight you have with your parents after you are eighteen. They'll yell at you and say "As long as you live under our roof..." just say "OK, I'll be gone in a week."
Since you can't work full time as a sixteen year old it will take you two or more months to save up enough to pay just one months rent for an apartment (unless you are some super entrepreneur). So start now.
I have a big cousin who is about twenty years old. He works at a Wal-Mart unloading trucks on the weekends and earns $600 for two days work. I have no idea why they pay that much. It would be a great job if you could get it.