Neither the P7700 or the G15 are pocket-able nor the 20x zoom you originally specified. Between the two, the G15 would be my choice. Powershot G series has always been a good performer. I was a big fan of the G's. Nikon may make great DSLRs but never really had their best effort placed into compacts.
Since you are relaxing your requirements....
* If you place your priority on zooms on a fixed lens compact, Panasonic FZ200 would be my choice. 25-600mm equiv lens and f/2.8 throughout the range.
* For a pocketable super zoom (with a compromise in other things) TZ series Panasonic is great. I have had a few in the series and happy but I wouldn't consider this a camera a good start for learning anything more than composition.
* If image quality in compact (sacrificing zoom) is your priority, I would highly go for the FujiFilm XF1. Big sensor, good controls, lens collapses into the body, pocketable, very unique EXR sensor for high ISO range and extended dynamic range Even better, the Sony X100 but the price is even higher. Even better... the FujiFilm X10 (or 20) but the price climbs way higher.
the suggestions can go on and on and on depending on your priorities.... people here can throw at yah 20 camera options to spin you around confused.
Heck, if you it just has to be smaller than a DSLR with the capabilities of a DSLR.... you might as well as go for micro 4/3rds system cameras. You then get interchangeable lenses, good sensor, over 10 bodies to choose from, large selection of lenses, and yet still smaller than DSLRs. Some of these micro 4/3rds cameras are actually smaller than some of the cameras being considered in this thread... and provide better platform to start learning.
As I said in the beginning, your search will be difficult if you don't order your priorities and relax some.
PS> Small mirrorless cameras IMO have rendered most high end P&S/compact and Bridge cameras obsolete in my opinion. The only high end P&S/compacts that are worth considering are the ones that combine, large sensor, good optics, and still are pocket-able. This includes the a for mentioned Sony and Fuji. As much as I liked the Powershot G's, this is the reason why I haven't purchased another since the G9. It gathered dust since acquiring a micro 4/3 camera.