Ok, I was kind of surprised that this hadn't been asked here before, but maybe it's too technical for this site. You guys decide.
I've heard lots of different stories about setting up Wine on Ubuntu, WineTricks, PlayOnLinux etc., but never a 'This is the best way to do it for Steam and Steam games' thread.
So has anyone had any real success getting their Steam games to run on Ubuntu through Wine or something similar? If so, could we get some specific steps?
There is no best way to do it, as such. Both Steam and Wine are ever-changing, and today's best method might break tomorrow*.
People have reported that Steam works with just standard Wine.
Install Wine 
from the Software Center (or click the shopping bag icon).
Right click the installer executable, go to Properties → Permissions and check Allow executing file as program, then right click it again and select Open with Wine Windows program loader.
If it turns out that doesn't work for you, you can try using PlayOnLinux.
Marco has detailed the instructions on how to get it working behind the link.
If it still doesn't work, you can give Crossover Games a go. Though it is number three on this list on purpose.
In general, it seems that Steam will work okay. You should try to run it using out-of-the-box Wine first. If you run into any trouble, feel free to ask a question under the wine tag at Ask Ubuntu.
Also see this site for Steam games that work with Wine (and, by extension, PlayOnLinux).
* Note: If you get it to work, updates to Wine will most probably never break it. But it may well be the Steam people are fighting off Wine compatibility. It may also be they are actively trying to make their program run well under Wine (games publishers have been known to do that).
In addition to Wine, WineTricks helps really much. It installs all the required drivers, libraries and frameworks into Wine.
Steam Games on Linux website contains Wine configuration and things you need to install (including WineTricks for some games) to play different Steam games.
I successfully played Alien Swarm and Half-Life 2 Deathmatch, though they don't seem to need any special configuration.
As of 15 Feb 2013, Steam is officially available on Linux!
Here's what the Valve Developer Community has to say on the subject.
One "easy" method of installing Steam is with Play On Linux. First download the latest version of PoL and install it. Then run it by typing: "playonlinux" in the console and click on "installer", select "jeu" or "game" and next, select steam, next, say yes for all next steps, and here it is, Steam is installed! Enjoy.
[For Wine, you] have to set up a working Wine installation first. [Then,] Download the installer, open a terminal and change to the download directory. Run wine start SteamInstall.msi and follow the instructions. After that Steam is installed in Wine's "virtual" Windows drive, usually ~/.wine/drive_c/Programs/Valve/Steam. Alternatively, you will need to use msiexec to run the Steam installer. Therefore, run wine msiexec \a SteamInstall.msi and follow the Steam installer instructions. Note: The login window doesn't have keyboard focus when starting up. You have to right-click into the login field first.
It also notes that if you run Compiz on your computer you will experience severe slowdown problems. It recommends to disable Compiz before launching Steam.
Steam works very well in plain Wine.
I actually paid for Crossover games since they provide commercial support and fix reported bugs in supported games extremely fast. Plus they push fixes back into Wine.
I use CodeWeaver's CrossOver to run games on Ubuntu. They offer paid support to get games to work.
You could try http://transgaming.com/ (Cedega). I did this in the past and it worked fine, but you have to pay for it - :\
