I have watch all of Dragon Ball, Dragon Ball Z, and that which is available of Dragon Ball Super, but still can't figure this out. What do the abbreviations for the different Super Saiyan transformations stand for in DBZ? For example, I've seen "SSGSS." Does this mean "Super Saiyan God Super Saiyan?" What are the abbreviations for Super Saiyan 1 and Super Saiyan 2?
Yes, you are right. SSGSS stands for Super Saiyan God Super Saiyan. I know it doesn't make any sense, but according to the info I gathered, "Super Saiyan God Super Saiyan, more commonly known as Super Saiyan Blue, is a form that can be accessed after absorbing the power of Super Saiyan God and then combining it with the Super Saiyan form."
For other forms like Super Saiyan 1 and Super Saiyan 2, no fancy abbreviations are used. They are generally known as SSJ/SS for Super Saiyan 1, SSJ2/SS2 for Super Saiyan 2, and so on. There are also other forms like USSJ (Ultra Super Saiyajin), which are sometimes confused with Super Saiyan 2 or 3, but note that they are different. You can find more details on this site also.
SSJ1 = Super Saiya Jin 1
SSJ2 = Super Saiya Jin 2
SSJ3 = Super Saiya Jin 3
SSG = Super Saiya God
SSGSS = Super Saiyan God Super Saiyan
SSJB/SSB = Super Saiya Jin Blue*
LSSJ = Legendary Super Saiya Jin
SSJR = Super Saiyan Jin Rage
*It's the last transformation with a new name. The new name was introduced in the manga because the fans were complaining the former name was too complicated.