Update from 10.5.8 to Lion with the Lion USB stick
Solution 1:
I know you can do a fresh install on a clean system, so you could try this: copy your existing install to an external drive (using Carbon Copy Cloner or SuperDuper!), wipe the internal drive, do a clean install of Lion, then use Migration Assistant to copy your old system onto the fresh Lion install.
Solution 2:
The Lion USB drive you bought works to upgrade a Mac with Snow Leopard (10.6) or you can of course use it to install Lion onto a bare (erased) drive.
Most people will make a back up (using the bootable USB drive to run Disk Utility) to an external drive or alternately - install Lion onto an external drive.
At that point you can hand migrate things or see if the Migration Assistant will help to move your documents and the Apps that are compatible with Lion to the new OS.
Since you just bought an Apple product, you get 90 days of phone support - so you can call AppleCare for help step by step if you prefer. The only catch would be if your Mac doesn't meet the minimum requirements for Lion in which case you might take it back and return it if you don't want to spend money on whatever needs updating to run Lion.
Solution 3:
Since the Apple Lion page points out that Snow Leopard is a requirement for the upgrade, the Apple Store employee should've pointed this out before you bought the Lion USB stick. If you're not willing to upgrade to Snow Leopard, @CajunLuke's answer would be helpful.
There are two paths to obtaining Snow Leopard. If you're currently a MobileMe user, this article notes that Apple is offering free upgrades to Snow Leopard to encourage user migration to iCloud. One other alternative is to purchase the Snow Leopard upgrade. Either way, install Snow Leopard first, update the OS, and then install upgrade to Lion.
Solution 4:
Are you sure your Mac meets the minimum system requirements?
- Macs with an Intel Core 2 Duo, Intel Core i3, Intel Core i5, Intel Core i7, or Xeon processor;
- At least 2GB of RAM; Mac OS X 10.6.6 or later (Mac OS X 10.6.8 is recommended);
- At least 7 GB of free hard drive space.
Any Mac with a Core Duo CPU will not run Mac OS X 10.7 Lion - basically any of the 2006-era first generation of ICBMs (Intel-chip based Mac).