Here's a rundown experience as a brand-new user.
I'm posting this on my Surface RT through my desktop's Opera 12. My home PC is Windows 7 Home, so it doesn't support Remote Desktop natively. TeamViewer is quite painless. I ended up downloading the QuickSupport version on my desktop (no installation or admin rights, just double-click and run) and then the app for RT through the RT app store.
The desktop application generates an encrypted ID and password, and you log in with that from the RT app (the desktop app must remain open for the connection to work). The desktop video resolution is forced to match the Surface RT's resolution so things don't look wonky. The desktop must remain on (you can use your Windows credentials to log in if needed), so I wouldn't use this on your target desktop if there are any privacy issues, like spouse/kids staying at home while you connect remotely.
There is a bit of lag in terms of mousing and scrolling. Opera is not touch friendly but my muscle memory went for the keyboard shortcuts anyway, and Photoshop CS5 was definitely laggy but doable in a pinch. I wouldn't try to game under these conditions. Lag isn't too bad while typing.
I took a couple screenshots on the RT, edited lightly in Paint, and uploaded to SkyDrive so I could access them on the desktop for this post.
Upon disconnecting, the desktop resolution returns to normal. After closing the non-installing desktop version of TeamViewer, you'll get an advertising popup or two, which is no big deal. Reopening TeamViewer QS, your ID stays the same but the password changes, so if you want to start a session later in the day, you'll have to run the desktop app first and write down the password for later use, and do NOT close the desktop app. All in all, a great solution I think.