To delete frames at the beginning you need to delete the keyframes from frame 1 to the frame before the keyframe you want to use as the new frame 1. You can not delete frame 1. You have to manually move the remaining frames (select all of the remaining frames and drag them until your new start frame is at frame 1).  

What you will end up with is empty frames on the end of your animation.

Hope this helps you a little:)