and should already be available on the phone.
Once you set it up it will store all photos taken to google's cloud storage for free if you accept the down sampling they do to reduce the photo size. In my experience unless you are doing professional photography the free version of the photo is perfectly acceptable.
You can choose to have it save in original format but it will over time require you to purchase more storage.
Either way once you do it it is pretty much set it and forget it. All photos taken will go automatically to google storage where you can access them from any device by signing into your google account.
Once the photos have gone to the cloud you can safely clear the phone storage with no worries of losing any thing.
Google photos is amazing BTW. You can search your photos in many many ways, You can for example search for trees and google will pull all pics that have trees in them. Or dogs and get all your photos with dogs in them. It is truly amazing. Not perfect but damn good and again amazing.
It is seriously the best and easiest way to store your photos safely with very little work on your end to make it happen.
Another little bonus you get with it is google will often put some fancy editing into your photos automatically. Creating some great collages or even stringing video together from a weekend trip to make a nice video with background music you can share with others.
They really have made it an amazing service and for your issues it would solve your problems completely with very little effort on your part.