When you install applications, their application files go in root. That means libraries, the program executable and global data, such as documentation, images, etc. User data, such as configuration and file data, goes into your home.
I would recommend a slightly larger root partition.
It isn't necessary to add a separate /home partition. If you choose to use BtrFS instead of Ext4, then it's not even recommendable. It's better to have a single root partition, since BtrFS can have subvolumes. This means you'll get both two separate filesystems, but do not have to have predetermined size limits for each. If you use BtrFS, then Ubuntu will automatically create a root and a home subvolume for you, so you don't have to worry about that at all.
BtrFS is fantastic stuff. For instance, it let's you jump back in time if something goes wrong. So, for instance, you can upgrade to the next version of Ubuntu, and if it doesn't work, you can undo the upgrade in a moment.
Be advised, that as BtrFS is very new, there is less documentation for it. But from an install perspective, it's easier. There's no doubt this is the future of Ubuntu.