I am not familiar with Gateway machines but this should work for most computers. You will have do some reading up. Yes it's probably a VGA (15 pin D-sub) the connector is often blue but can be any colour)
If your mouse/k-b etc are wireless they may
need fresh batteries.
It may be the graphics facilities on your computer are u/s. It's quite awkward to solve this problem because you can't see what you are doing. You might try drivers for the monitor you are using (do not pay for these and be careful to void websites which are sources of adware or other malware) though this is unlikely as basic graphics are supplied in the BIOS
Try "gateway" support, or the "gateway" or other computer forums.
Read this http://www.pcguide.com/vb/showthread.php?70528-Upgrading-Video-Card-on-Gateway-GM5420
If there is a peripheral graphics card and your machine has on board graphics / video enter the BIOS (read up) and choose on - board video.
Try a graphics card (or on-board, or a different one known to work - see one of the many articles on how to set up a graphics card) Borrow one.
Essentially my course of action is try an alternative graphics facility to the one you have at the time of your post.
(Ihave just looked at the specification for this model at http://www.cnet.com/products/gateway-gm5420-core-2-duo-e6300-1-86-ghz-2-gb-320-gb/specs/
and it has on board Intel graphics default. Assuming the hasn't been an graphics 'upgrade' on your PC
You might find this webpage helpful (it has a troubleshooting section in the service manual)