I'm having trouble finding a jailbreaking program that completely works for me.

​Is it possible to Jailbreak the New iPad? Mine is running with the iOS 5.1.1 If yes, which is the best program available online to do it?

I tried the Jailbroken Programs "absinthe-win-2.0.4" and "redsn0w_win_0.9.14b2", I followed the instructions step by step, so far I'm able the use Cydia perfectly without any problems at all. But when I try to back-up, I get problems with iTunes. When I connect the iPad and back it up, if I added an app using iTunes on the computer all works fine. But if I use the Apple Store on the iPad when back it up all the apps I added get erase (and they aren't save on my iTunes).

Greetings.
You might try transferring purchases in iTunes. Highlight your iPad and then right-click or control-click (Mac only) and choose "Transfer Purchases", or click File then Transfer Purchases. This will copy the rules you Purchased on your iPad to your computer.
iTunes shouldn't be erasing the new apps that you have on your iPad.
You might want to check that your iTunes account on your computer is logged in with the same username as your iPad. It may be that your computer is restoring your iPad back to an earlier sync from a different account.

Sent from my iPad using iPF - Peace