Yes, you can download SC2 through the Battle.net client without having purchased it and play certain modes.
It seems that you can play...
A handful of campaign missions, but not the full campaigns as they each require buying their own respective expansion pack.
Non-ranked multiplayer skirmishes with friends and AI.
Co-op scenarios with a handful of heroes.
Any custom maps in the arcade, which can vary from things like DoTA (which was originally a custom map for WC3) to Tower Defense, really people go wild with custom maps for Blizzard games and SC2 is no exception.
It also seems that regardless of what expansions you own, you can set which 'expansion level' you are using. So you can either see only the original units from the first game, Wings of Liberty, or the entire progression of multiplayer changes through to Legacy of the Void.
I can't say exactly how much of this is fully free and how much is available because I own the original game, but I can access all of this without owning the 2nd or 3rd expansions.