I have always been able to see the taskbar while in Minecraft full screen and have been able to easily change to the browser from the game and back again. I accidentally pressed something and now the taskbar is gone. I can still use the windows key to get to the browser, but it makes the computer really slow for some reason. I want to be able to have Minecraft take up the full screen, but still be able to see the taskbar at the bottom.
Here is a visual: 
Are you sure you didn't use maximized windowed mode? Fullscreen usually tends to take 100% of your screen, hides the taskbar and everything. Hit F11, Then the maximize button on top right of the window.
You probably hit F11 by accident as that is Minecraft's shortcut for full screen mode. In proper full screen mode you're not supposed to be able to see your taskbar. You were probably using it as a maximized window beforehand. To fix this all you must do is hit F11 again (or turn off full screen in game settings) and make the window maximized by pressing the square in the top right corner of the minecraft window.
