As far as i can remember, when you start the game (or when you first exit the sewers/tutorial section of the game) you'll receive a lot of pop-up windows with various information about each extra piece that is included in the GOTY edition. I've only own the retail versions of Oblivion (regular & GOTY), but i can't see why the above wouldn't apply to the GOG version as well.

When you start the launcher, there should be an option to choose which plug-ins are active. If you have the extras available, you will see a list of each plug-in; to enable each plug-in, check the boxes. (They're not enabled by default.)

(Note that enabling more plug-ins may introduce bugs, so you might want to only enable the ones you are interested in. You can always enable more plug-ins later.)

If you're talking about the GOG version and you can launch the game after the installation is complete, then you have the full GoTY. As I haven't installed Oblivion for a while, I don't remember whether or not the add-ons are activated automatically or not, but I'm pretty sure at least Shivering Isles is.
You can check it by launching Oblivion, and then clicking "Data Files". This produces a list of all currently installed add-ons. If an entry has a ticked check-box, it is activated, otherwise it is not.

Thanks for the responses everyone. I really appreciate it.

THANK YOU!