How long have you had it?
Gear cables will stretch over time (especially in the first few months) and that can cause the sort of problems your having with the rear derailleur.
The chain coming off the front when you move to the big ring just needs the limit screw adjusted.
Here is a guide for adjusting the front derailleur. Ignore the bits about setting the height and rotation, just scroll down to the bit about setting the limit screws. It's quite as easy thing to do.
Gear cables will stretch over time (especially in the first few months) and that can cause the sort of problems your having with the rear derailleur.
The chain coming off the front when you move to the big ring just needs the limit screw adjusted.
Here is a guide for adjusting the front derailleur. Ignore the bits about setting the height and rotation, just scroll down to the bit about setting the limit screws. It's quite as easy thing to do.