I want to reinstall Ubuntu on a new drive and keep my personal data on the old drive

You can keep the hdd as storage and install Ubuntu to the SSD.

First, to be safe, you can unplug the HDD and just do the installation for the SSD while the HDD is unplugged.

Then, make sure the SSD is the first boot device in your BIOS settings to prevent the HDD from booting your old installation.

After installation, plug in both drives and when you boot Ubuntu, you will be able to access the HDD as a storage device.

You can either leave your old Ubuntu installed on the HDD and it won't make a difference or you can delete the old system files individually. Just be careful not to delete your movie files (and probably make a backup first just in case). However, the old system files really won't take up much space.

If you have any trouble or if you're about something let me know. Thanks!

That's perfectly possible:

• as mentioned in the other answer for safety unplug the old drive,
• install new drive
• install Ubuntu on the new drive, make sure it works, update it
• advanced option is to use LVM during installation - it will give you most flexibility and your skills will have grown
• this question discusses advantages and disadvantages and gives good overview: What is LVM and what is it used for?
• once you're satisfied with your new Ubuntu installation plug-in old drive and mount selected partitions
• if your old install was using LVM you may need to import volume group before you can see the old partitions (logical volumes in LVM terminology)
• details here: http://tldp.org/HOWTO/LVM-HOWTO/
• after you can see the partitions and you are able to mount them manually you can add entries in /etc/fstab so selected partitions get mounted when the system boots up
• here is the detailed guide: https://help.ubuntu.com/community/Fstab

For example my data drive is mounted over /home/user/data

This way your old drive is accessible in this directory while everything else resides on the new one.

If during the process you do not understand anything or something does not work as expected I encourage you to ask separate questions or enhance this one.

While fiddling with the old drive is all too easy to break the drive structures and loose the data.

Later you can also delete old Ubuntu installation from the 2nd drive and reclaim the space - the exact process depends on how the old Ubuntu was installed and whether you used LVM.

Once you have your both drives up and running describe the install and post the output of sudo fdisk -l command so we can help you reclaim the space.