How exactly can Time Machine help me?

Here are some instructions for opening TM on a different Mac:

1. Browse to the network drive in Finder
2. Double-click on the sparse bundle file for the computer you want to access.
3. Click on the "Time Machine Backups" disk image in Finder (added to the left-hand pane by step 2 above).
4. Open the backups.backupsdb folder inside this disk image.
5. Continue as per previous comments.

But instead of this, I recommend getting something like Carbon Copy Cloner (CCC) or SuperDuper to clone your drive to a separate external disk. Not only do you have a bootable copy, but you don't have to mess around with TM at all and can access the files as you expect on the new/different Mac.

 

Yes. Plug in the Time Machine backup. It'll literally show up like an external hard-drive. Couldn't be simpler. You don't need to write the backup to a new machine.

You may have issues writing to certain folders on the backup drive, but you'll definitely be able to see all of the data.

Restoring a Time Machine backup will restore everything, including the OS. Restoring an older OS to a new computer is thoroughly, thoroughly unrecommended.

A workaround would be to use Migration Assistant when you've set up your new computer, to copy data/applications back from the Time Machine backup. Otherwise you can just drag-and-drop the data you want.