As you noted, you'll be running the
DBCC CHECKDB command (note that
DBCC just means "Database Console Command", whereas
CHECKDB is the command it is executing).
This will be sufficient to ensure that the databases have survived the restore process.
For complete peace of mind, you can issue this command on each database:
DBCC CHECKDB WITH EXTENDED_LOGICAL_CHECKS, NO_INFOMSGS;
EXTENDED_LOGICAL_CHECKS will do extended checks on "indexed views, XML indexes, and spatial indexes, where present" (per the documentation);
NO_INFOMSGS returns no results if no errors were detected.
Incidentally, updating the compatibility isn't a required step. The databases can stay at the 100 compatibility level until you're ready to take advantage of the newer features of 2012 that require a newer compatibility level.
If you perform your backups with the CHECKSUM option, you can do an additional verification check during the database restore for added peace of mind.