Putting the SD card into the phone should allow you to format the card so it can accept data directly from the phone. A file manager app may prove useful for you - use it to move files from phone storage to the SD card. Set your apps like photos and music to save to the SD card, rather than the phone's internal memory. Move your photos, videos, and music to the SD card to free up space on the phone for apps.
As far as the cable only charging when plugged into the computer - try pulling down the notification bar on the phone to get options as to what happens - it defaults to charge only, but you can change that to transfer data.