Please uninstall AVG from the control panel and using their removal tool. You can not run 2 security programs at the same time, not 2 real time security programs like N360 and AVG. Of course AVG won't open because it can't install properly with N360 installed. You may have messed up your install of N360 by doing that also..
Did you attempt to install N360 on an already infected computer? Installing it on an infected machine will also preven it from working correctly. What version of N360 do you have now? What operating system and what service pack and is it 32 or 64 bit? The program may be waiting for it to do it's first idle system scan on it own.
Let us know the version of Norton 360, you can find it from Help & Support > About. Try to restart your computer in Safe mode, double-click Norton icon and proceed with running a full system scan. If it detects any threats, fix those and restart back to normal mode. Run a comprehensive scan after this.