What you see in the Gallery is for the most part shortcuts to the actual files. You can go to the app manager and clear the cache and settings and nothing will be lost but you will save some space and the Gallery will rebuild the cache as needed.
You don't need a file manager if you don't want to go that route. You have an app called "my files" open it and select "all files", "device storage", and select the file that you want to move (might be "pictures" or "dcim") you can then select the files and from the drop down box in upper right select "move to". Then select an SD destination and move. It's very quick and easy on this phone! Way better than previous androids. Hope this helps.