One solution is to use the command line in OS X and chmod the directories to 777 (or 755 if you just want read permissions).
Open the Terminal in OS X. Type
ls and you should see a list of your folders like Music, Documents, etc. Now say you want to change the permissions on Music so you can read and write to it in Ubuntu. Type:
chmod -R 777 Music
-R is there to recursively change permissions on everything contained within Music. You would use
755 if you are fine just accessing Music, but didn't want to be able to add or delete files to the Music directory from Ubuntu.
That works for reading and accessing them. You will not be able to write to them however unless you disable journaling on the OS X partition, which you can do either from the command line or in Disk Utility (click on the OS X partition, then hold down option key while clicking "File" menu to enable "disable journaling" option). The reason for this limitation is explained in this other thread.
Actually there is a way to keep journaling on and write to that partition, but you would need to install
hfsprogs, see this for further details.
If you're feeling overwhelmed, I guess I would say there's no urgent need for the journaling. It's good to have it, but it wasn't even introduced into the consumer version of OS X until 10.3. Were Mac users using 10.2 in great danger? If you experience frequent unexpected shutdowns or power losses, then I'd definitely turn it on though.
Also, you might wonder why people bother changing the UID instead of just changing permissions. This is because changing permissions allows other users on your computer to now access those folders. If you're on a personal laptop where you're the only user, this shouldn't be an issue (unless you let other people log in remotely for some reason). If you do let other (untrusted) persons use your computer and you don't want them to access the information in those folders with the changed permissions, you should consider the UID route.