My iPhone is also synced with my MacBook and I have yet to even start thinking about transferring my data and making it sync with my new iMac.
I believe if you use Migration Assistant or transfer the backup files manually to the new computer, you will be asked to resync the device with the new computer and it should behave as normally since you're restoring from the backup files. Someone correct me if that's not how it goes.
I believe there's also a way to edit iTunes on your new computer to have the same ID as your old computer, fooling your device into syncing natively without doing the restore from backup thing. I haven't looked into it in detail as it's too much of a headache right now for me to migrate my iPhone to the new computer so this paragraph is just me deducing on how it works from reading a guide I found for about 27 seconds.
And hijacking the thread, do the backup files contain our saved games information from games or is that lost in restoring from backup?