Android in all its wisdom has prevented adding or transferring data to SD. You can tether to a pc and transfer items. Connect to a PC via a data cable, make sure the device is unlocked and if applicable it is set to data transfer, some devices don't have this option and will transfer data if connected, open the My Computer folder on your pc and navigate to your device's internal memory, now open another folder and navigate to My Computer and then to your device's SD card. With the two windows open you can freely move items back and forth. If these are pics that you've taken you'll find them in your DCIM folder. If they are from other slices you will have to look for them yourself as I can't tell you specifics about unknown information.
Keepitreal makes a good point it would be prudent to backup your data to a cloud service. The easiest way to do this would be to use one of Google's services since Android is a Google product. There are 2 primary devices by Google for backing up data and both are simple to setup but for simplicity there is one that I prefer which is Google Drive. Drive I think makes sharing pics and things easier and you can use it for far more than just pics and vids. G+ is good for backing up pics and vids but only pics and vids and it can be a pain to simply view your photos from other devices because it touts itself as a social media network. More about that later.
If your device didn't come pre-installed with Drive look for it at the Play Store. Once find or installed open the app, you'll have to agree to some terms and conditions and select a Google account. Once all that is complete you should be sitting on a mostly blank screen with My drive across the top. Open the menu, the 3 lines on the left of the screen, scroll to settings, there you'll find a slider for Auto Add. Tick that and then down to Manage backups. There you can see backup sources and confirm that image data will be backed up.
Drive is simple to share a little, or a lot from. Simply press and hold the item or folder you want to share and pick how you'd like to share it. For instance share a whole folder with your family by sending them a link. In the future any items added will automatically be shared to anyone with the link. Share one specific document with a friend the same way. Or add their gmail account and they can have a permanent share item they can view from their Drive app. In turn they can just as easily share items with you. Also you can view and manage your items from any device you can log into Google from.
G+ is similarly easy to setup. I don't like this one because it's difficult to view your items from other devices, and it tries to be a social networking thing.
Using either of these services also makes viewing your photos simple, even after you've removed the originals from your device. In your app drawer you'll find an app called Pictures. This combines all your photos and videos in one place, even those from Drive and G+.
Before I forget moving forward you can select in your camera app to automatically save original photos and videos to your new SD on most devices. Look for storage and select external in your camera settings.