The obvious gallery to start with is the Tate Modern. The nearest tube station on the south bank is Southwark but most people find it easier to travel to Blackfriars and walk over the Millenium Bridge.
http://www.tate.org.uk/modern/
The much-praised Estorick Collection isn't free but, at just �3.50, it's not as expensive as many London attractions.
(Tube: Highbury & Islington)
http://www.estorickcollection.com/
If your interest in modern art extends to photography, you might like to visit one of my favorite free places in London, the Photographers' Gallery. It's housed in two separate buildings close to each other. (Make sure you visit both). The nearest tube station is Leicester Square.
http://www.photonet.org.uk/index.php?latest
For further suggestions, I recommend investing in a copy of London's premier listings magazine, Time Out. It's available from every newsagent in London. You can also get it at most station bookstalls on main lines to London and from larger branches of W H Smith throughout much of the country.
Chris