Ubuntu – Change installation path / move programs

I would argue for a single / partition in the SSD (60GB, as you have envisioned) to keep everything Ubuntu, including the /home folder.

A separate /home partition is useful:

(1) when one plans to try out different distributions of Linux (also known as "distros")

or

(2) when one may want to re-install Ubuntu without destroying the personal files and settings in /home.

The second reason is a bit less important now, as newer versions of Ubuntu installation process will detect existing Ubuntu and offer to preserve the /home folder when installing over.

On the other hand, if you keep /home as a folder of / you won't have to worry about how much to allocate to / and how much to allocate to /home. If your program files (games) take some more space, all you have to do is move your personal data to the HDD.

By now you must be thinking why not put /home in the HDD?

The only reason I prefer /home to be in the SDD is because /home stores not only your personal data, but it also stores your personal settings for all programs. This includes changes you make to the desktop or any other program settings, your browser bookmarks, etc. When you load a program from SSD but the personal settings are in HDD, it slows loading that program a tiny bit. Keeping / and /home both in the SSD will get you the fastest load time.

Keep large data such as music, video, saved games, etc. in the HDD

The folders that contain large amount of data, such as Video, can be in the NTFS partition with symbolic links to /home. This is explained in some detail at: Windows Ubuntu dual boot - Share files between OS

Other large sub-folders of /home can be partitions in the HDD. For example, if you install a lot of Windows programs and games in WINE, the hidden folder .wine will grow large. You can create an ext4 partition in the HDD and mount it at /home/[your user name]/.wine (Unlike Windows, a partition in Linux can be mounted as a sub-folder.) Steam games take up a lot of space. Your steam games are in /home/[your user name]/.local/share/Steam. See where are Steam games installed? You can create a separate Steam (ext4) partition in the HDD if you want. These will require a one-time edit of the file /etc/fstab. See Mount a partition on login? for more on how to mount a partition at login.

The reason I recommend .wine to be an ext4 partition rather than a folder in the NTFS partition is because NTFS cannot preserve ownership and file permissions used by Ubuntu that is needed by the settings and other files stored in the .wine folder by WINE.

Between system linked folders in the NTFS partition and ext4 partitions mounted as sub-folders of /home you can keep all the large files in the HDD while keeping the system fast, taking advantage of the SSD.

If you still run out of space in the / partition in SSD due to large games and other programs, you can create an ext4 partition and move /usr there. However, this will require some planning, careful thoughts and execution. Ask a separate question if and when you need to go on that route.

As far as installing programs in / versus the /home see this excellent explanation of the process and its limitations: Software installed on root partition or on home partition In short, programs installed via the Ubuntu Software Center, or the command apt-get will be installed in /. Only "static-binary" compiled programs may be installed (rather copied) in sub-folders of /home and expected to work.

More references:

Partitions for Ubuntu and Windows 7 dual boot on SSD + HDD setup

Setup for dual disk (SSD+HDD) with /home partition

Do I need Intel Smart Response when installing Ubuntu?

Move or Install Apps on HDD not SSD

You don't have to do that!

As you have read elsewhere the best answer is, don't mess with the system folders. You can move the /home folder to a dedicated partition in the HDD (your sdb) using the official help page However, I recommend keeping the /home as well as /home/[upir user ID] in the SSD as all your personal configuration files are there. Keeping the personal configuration in the SSD keeps the system snappy.

Steam

Steam games take up a lot of space. However, it does not keep the games in the system folders, such as /usr. Your steam games are in /home/[your user name]/.local/share/Steam. See Where are Steam games installed?

See Installing programs in root vs home partitions? for more on the how to use an SSD and HDD.

Follow the instructions in Do I need Intel Smart Response when installing Ubuntu? to move some of your large personal folders like ~/.local/share/Steam to your hard drive and create symbolic links to the SSD.

I recommend creating an ext4 partition in sdb so that file and folder permissions used by Ubuntu can be preserved. These will be lost if you dump the folders in a NTFS partition. See I want to reformat my home partition from ext3 to ntfs

Also see How to save applications for re-install or upgrade? for why moving some apps to a different partition and keeping the default apps in their original place is a bad idea.

Hope this helps