Using free space from a drive (sda) onto another (sdb)

You can't move (free) space from one drive to another but you can still make the free space available to your Ubuntu system.

Method 1: Use plain partitions

This is the more simple albeit far less flexible variant. The basic idea is to:

1. Decide which data you want to store on or move into the now free space.

   • You could create a new volume and “mount” it at /media/pehuen/my-data or /mnt/my-data or pretty much wherever you like and store some of your files and data there.

   • You could have all of /home including your (and other users') personal files and data in a separate partition. For this one refer to the excellent “How can I move my /home directory to another partition if it's already part of the / partition?”.

   • In a similar fashion you can move all of /usr, /opt or /var onto a separate partition (though that's a little more complicated to manage if you ever run into boot issues).

   • You can choose more than one or all of the above.

2. Create one or more new partitions in the free space.

3. Depending on your decision on step 1, move the affected files to the new partition(s).

4. Configure your system to “mount” the new partition(s) at boot.

Method 2: Use logical volumes

This is more complicated and requires more work and is mostly for advanced users. Logical volumes let you aggregate multiple physical partitions/volumes into one or more logical volumes that span one or more (or all) of the physical volumes. This allows you to add or remove physical storage as needed while you can grow and shrink existing or add new logical volumes very flexibly.

In theory you can have your entire Ubuntu installation including all system files and personal data inside a single logical volume that spans thousands of physical drives. A little additional work is required to get Grub to boot Ubuntu from a logical volume though.

Conclusion

There are two basic methods how you can integrate the free space from a different disk into your Ubuntu setup. They provide different degrees of flexibility at the cost of a great deal of complexity in setup and maintenance. Both require you to make a decision how you want to organise your data.

If you made your decision and laid out a basic, abstract action plan, you're welcome to open a new question on how to flesh it out and/or implement it. Send me a notification in the comments if you want to draw my attention to the new question. Please don't extend your current question with such a broad follow-up question since it would invalidate previous answers and make things more difficult to follow for future readers.