How to backup photos from iPhone
Google Photos lets you store unlimited photos in high quality on their servers, just by downloading the Google Photos app. The app works great, and I've found it to be reliable. It gets 5 stars in the iOS App Store (which is really quite an achievement) and has nice extra features in the Assistant section, like auto-making videos from your photos/videos. Once your photos and videos are backed up to Google Photos, you can view them on phones, tablets, and computers, just by signing in to your account.

You can also spend $1 a month to purchase 50 gigabytes of iCloud storage, and turning on iCloud Photo Library lets you back up all of your photos similarly (though I've found this to be slightly less reliable than Google's solution). 

Read more about backing up your photos with this New York Times article.