The Series 2 characters — the ones from Spyro's Adventure that have been redesigned for Giants — can be played on both Spyro's Adventure and Giants.
Here's a list of the Series 2 characters that will work on both games:
You can play on different gaming consoles, too. This means that a Skylander can start leveling up on a Wii, and then go help another Portal Master on Xbox 360, then play on a PlayStation 3 or even a 3DS. This works no matter what console is used.
If a Skylander reaches a level higher than 10, it will be level 10 in Spyro’s Adventure. If a Series 2 character unlocks their Wow Pow ability in Giants, please note that it will not be available in Spyro’s Adventure.