Download the starter version. All this requires is you to make a BattleNet 2.0 account (free, does not even need any games attached).
The SC2 starter edition is the latest version of the game. You will be able to enjoy all arcade (custom) maps made for SC2 and can partake in some melee activities as long as you are in a party with someone who owns the game. Although you can play all Arcade maps, StarCraft II Editor requires you to upgrade to the full game.
Upgrading to the full game is done either by purchasing from the Blizzard online store directly (it will attach to your account as soon as processing completes which is usually 1-3 days) or by attaching a CD KEY located in the case of physical copies. Various third party sites offer you the CD KEY directly usually at a discount however these may be subject to different consumer rights or even be fraudulent so use at your own risk. In all cases some kind of monitory transaction is necessary.
If you already own WoL you will also have access to all Arcade maps. You will be able to use the StarCraft II Editor as well. However all HotS content in the StarCraft II editor will be blocked out until you upgrade to HotS. It is important to note that with the editor the blocked content is only the HotS data mods, all version 2.0 and later features will be accessible for you to use. Additionally HotS melee is at Starter level only so you will still need to party with someone who owns HotS however you will have full access to WoL melee.
For people who are unfamiliar with Blizzard's game distribution model, there is only a single distribution of the game. No mater which version of the game you have a licence for (starter which is free, or WoL or HotS, or their collector editions) you will still be using the same client version.
The versions are as follows.
Wings of Liberty : 1 - Obsolete, please update when possible.
Heart of the Swarm : 2 - Current version at time of post.
Legacy of the Void : 3 - Future version reserved for the final currently scheduled expansion pack.
Currently all StarCraft II clients should be at version 2. This will be done automatically when you try and log in either using the Blizzard launcher program or the program executable directly. Do note that Heart of the Swarm will require approximately 8GB of data to be downloaded which may take considerable time and could conflict with your internet policy.