In the main menu of the game, click Battle.net. From there you can log in or create an account.
Click the Play Game button in the top left corner once logged in to play a game against a randomly (hopefully skill-matched) opponent. Note that people have been playing Wacraft 3 for a really long time now and have gotten pretty good at it.
You can also create or join a custom game. There are loads of fun mods people play online.
Once you get WC3 working, I've got to put in a shameless plug for Swat: Aftermath, one of the greatest custom games ever created for TFT. Just head over to the USeast channel SWAT and ask anyone there to show you the game. Great team based play.