Every Xbox comes with a hard drive so you shouldn't have to think about saving games elsewhere. However, there are memory units available for both the Xbox and the Xbox 360 which should allow one to save games on them. However, they are low capacity units and are fairly expensive.
While a lot of Xbox games do work with the Xbox 360, NOT ALL games work. Games that work have been identified by Microsoft and you could probably google to get a list of Xbox games compatible with the Xbox 360.