When you set up your Ubuntu installation, you mentioned that you placed the root directory (/) on the SSD, and the home directory (/home) on the HDD. This is a good start. I actually recommend placing root on your SSD, as well as home, and then create a directory such as /mnt/data which is where you would put all of the data you wanted on your HDD. With this setup, I can easily symlink stuff on the HDD to my SSD, such as video games and movies, while keeping my system as fast as possible.
I'm not quite sure why you want what you're asking for, as I'd personally want system stuff to be on the SSD to increase speed, while putting my less-important stuff on the HDD, however you can accomplish this much like how you set up your installation in the first place.
Instead of placing the home directory on the HDD, you could mount /usr/bin or even /usr. This is the typical place you'd find apt installing binaries.
Before you take the time to do this, I'd highly recommend using baobab or something like it to figure out 100% where you're losing space.
If you'd like to separate both /home and /usr, you'd just have to split the HDD into two separate partitions.
Other Ideas
As Barafu Albino added:
- Use
apt-get clean regularly to remove unneeded packages and apt downloads
- Think about using BTRFS with high compression as your filesystem
- Refrain from multilib and debug (dbg) packages
Edit
10GB is probably very small for a desktop computer. The only time I've seen something that small is when someone really breaks out all of the other mount points onto HDDs and stuff (maybe 20 ys ago) or on an embeded machine or something like a raspberry pi.