Operating systems and software will frequently need to access multiple files simultaneously, with files located in different parts of the disk. Mechanical spinning hard drives perform very poorly with these "random reads" and SSDs excel at random reads since they do not have to physically seek to data on the disk.
So it would be wise to install your operating system and frequently used software on the fastest drive. It will result in a significant improvement of overall performance compared to installing the OS and software to a mechanical spinning hard drive.
The performance gap narrows with static files like music, videos, pictures, and documents, as these are generally loaded on demand, one at a time, in a predictable order. Often, the difference is imperceptible during normal operation, even though read/write speeds are generally faster on the SSD. The exception would be when copying or writing a large number of files at one time. But if you're not doing that often, put them wherever you like.
It's not wise to over partition especially if you are new to Ubuntu. You can use the guided installation to "Erase disk and install Ubuntu". You don't need to create any extra partitions. You do not need to manually create a boot partition. You do not need a swap partition. You do not need a separate home partition. Some advanced/veteran Linux users like to create a separate home partition as a matter of convention and also because it can be easier to format your system partition without deleting your home folder. But it is not quite as easy as it seems to reinstall a new system and just attach the former home partition. /home contains a lot of configs that might not play well with a new installation, especially if it's a different version, flavor, or distro. Creating partitions solely for the purpose of organization is unwise, as folders serve the same purpose and do not place artificial limitations on your space.
If Ubuntu will be your only operating system, then install Ubuntu to the SSD. It would be easier to just use the whole drive, and I suggest that you do. Ubuntu itself will not take up the whole space but you can store frequently used data in folders on this drive to keep things organized.
