I want to update my phone

To update to the most recent, compatible version of iOS 12.5.7, please see: Update your iPhone, iPad, or iPod touch - Apple Support

And: If your iPhone, iPad, or iPod touch won't update - Apple Support

If the above Apple articles do not help, please post back with relevant information. 

For iOS 15, you will want to acquire an iPhone 6s or iPhone 7, and—for current iOS 16—you will want to acquire an iPhone 8 or newer.

There are no hardware upgrades for an iPhone 5s, or any other iPhone.

Options are a newer (used, refurb) or new iPhone.