I'm attempting to upgrade to OS X 10.7 from 10.6.8. (10.7 is the highest I can go on my MacBook) I purchased it online and waited 3 days for the licenses. Anyway, following all the instructions and attempting to download using the App Store. Aaand...nothing happens. Well, not NOTHING but nothing useful. All day yesterday it showed a little spinning image like it was getting ready to download then nothing. Finally last night it actually showed that it was downloading something but from the looks of it, it was going to take about 2 days to finish. It was about a third complete last night before I went to sleep then this morning it was all gone, nothing. Today I'm attempting again and restarted and there is a nice little icon on the Dock telling me that the download is Paused. It will not Resume and it will not go away.
Try going to the App Store/Purchases tab and see if it is there. If so, there should be a Resume button there.
If you get it downloaded, I recommend you make a copy of the installer and move it out of your Applications folder. The installer self-destructs. The copy will keep you from having to download the installer again. You can make a bootable USB stick to install using this free program.