First. iCloud Storage and the Internal Storage on your iPhone are two different things. You will not get more space on your iPhone, even if you buy more space in iCloud.
For example, apps and the data they contain will still fill your iPhone's internal memory. Music you download will fill up your iPhone, messages will fill up, photos will fill up, etc.
iCloud should be seen more as a synchronization/back up service, more than storage, if you ignore iCloud Drive which acts as a storage space in the cloud for files, just like Dropbox or Google Drive. However, files in iCloud Drive will still fill up if they are downloaded locally.
As for your photos, you can activate Optimize iPhone Storage. With that function activated, your photos will be saved in a low space-saving quality locally on your iPhone, and the original quality will remain in iCloud - until you need to view/share the photo, then the Photo app will automatically download the original quality . This feature should be enabled by default, but you can check for yourself here, go to:
Settings -> Apple ID (Your name) -> iCloud -> Photos.
Make sure Optimize iPhone Storage is enabled.
Beyond that, there is only 1 way. Clean up. Go to:
Settings -> General -> iPhone Storage
There you can see what is taking up space and choose to remove What you don't use.