I tried to edit paradd0x's answer above to include the option of using Ctrl+Alt+Home in conjunction with unchecking Display the connection bar when I use the full screen. This keeps the connection bar out of the way until you specifically summon it, which seems the perfect solution to Some Noob Student's original quandary, but the mods inexplicably rejected it. (Incidentally, overly strict crap like this is why I don't use these sites as much as I used to. Why would you reject an edit that takes the top voted answer and makes it even more clear and applicable to the question?)
An alternative to consider:
I use Chrome, which has the same "problem" with the tabs. I'm also a long-time Winamp user and use it in Windowshade mode at the top of my screen, with Always on top set to true. When Chrome isn't maximized it leaves just enough room for Winamp to hover above the tops of the tabs. This combined with the fact that maximizing any web browser on a 24" monitor (all my monitors are 24") is a waste of horizontal space, means that I literally never maximize Chrome.
As an aside, I use WinSplit Revolution which is a (free) utility that places/sizes the focused program wherever you want. For instance, pressing Ctrl+Alt+Numpad5 will position Chrome (or whatever's focused) in the center of the screen at 66% width and 100% height, which is my preferred browser size on a 24" monitor. Additional presses of the same keys cycle through different placements/sizing, one of which is 100% width, 100% height, but not maximized -- this would allow you to use your browser "full screen" if that's how you prefer, but still have the tabs pushed down away from the top of the screen. WinSplit is the single most used/useful utility I've found in my 15 years of working with computers.
Anyway... where this applies to Remote Desktop is Winamp will actually show over the top of even a full screen Remote Desktop session. When I want to switch back to my local machine, I simply click Winamp once and the taskbar from my local machine appears -- voilĂ !. I mainly RDP into my workstation, so I prefer not to minimize the remote session. Running local programs at less-than-full screen width leaves portions of the remote session visible in the background (not to mention completely visible on my second monitor) allowing me to keep tabs on incoming email, new IMs that pop up, etc. I'm so used to the convenience of this that I run Winamp even when I don't have it playing any music.
CTRL + NUMBERto change tab but if you don't have focus already it doesn't work. – Amicable Jun 25 '12 at 9:13