Download the photos from your phone onto your computer. Flickr or Google Photos both have automatic uploads from iOS so that’s not a bad shout either, but not sure if there is a quick way to delete the old photos and personally I like to keep my photos local as well as in the cloud.
In iOS 8.something they introduced “iCloud Photo Library” which is cloud storage for your photos. With this came the option to “optimise iPhone storage” whereby the phone only keeps thumbnails of your photos and downloads them from iCloud when you want to view them. In theory popular and recent photos are cached locally on your phone.
If you’re on iOS 7 you won’t have the option of iCloud Photo Library and, perhaps more importantly, you will soon use up that free 5Gb and need to pay for more iCloud storage for photos if you enable it.
My Photo Stream and Photo Sharing are something different, with the former only keeping your last 1000 photos in iCloud.
Apple and photo management is a confusing mess, but their website is as good a place as any to make (some) sense of it.