How do I upgrade the storage on my iPhone?

You cannot upgrade your storage on your iPhone. You do have the option to use iCloud with the Optimized Storage option along with iCloud Photos to store some of your data in the cloud. When using iCloud Photos with Optimized Storage, only the low quality photos are saved on your device until needed to save space.

As for your Home Button, this will need to be repaired by Apple. Depending on the cost, you may want to get an iPhone with more storage to solve both problems at once.

iPhone Repair & Service - Apple Support

You can't upgrade the local storage (device storage) on your iPhone. Whatever amount of storage it had when it left the factory is the amount of storage it will have for the remainder of its days.

You can rent iCloud storage on Apple's servers, for storing iPhone backups, and for synchronizing data (various categories) with other devices. As a special case, if you have iCloud Photos, you can set things up so that your phone has permission to reduce the quality of some of its local copies of your photos and videos when you start running out of device storage. The idea there is that the full-size copies are still around in iCloud from where the phone can download them later, on an on-demand basis.

What's the difference between device storage and iCloud storage? - Apple Support

Set up and use iCloud Photos - Apple Support

However, if your home button is broken, you might have more immediate problems. See the Service and Repair section of

iPhone - Official Apple Support

A faulty home button can probably be repaired (contact Apple Support), but the internal storage is fixed and forever whatever the iPhone was originally shipped with.

Options here are offloading contents elsewhere (such as to iCloud), or replacing this iPhone with a newer or new iPhone and preferably with capacity adequate for your current and likely future requirements.