My S5 will not let me add anymore apps to my phone. How can I increase the memory?
You can't. However you can move pictures and media to the SD card and clear the temp cache for the applications.
You can get a MicroSD card and move your content to the SD card to free up space on your phone. The S5 will support up to 128GB of expandable storage according to the tech specs of the device. You can get MicroSD cards from Verizon stores, Walmart or anywhere else that sells electronic equipment. They're fairly inexpensive depending on the storage size of the card and brand you get.
Alternatively, you could cloud your content using Verizon Cloud, Dropbox, or Google Drive to free up the storage from your phone so that you'll be able to download applications again.