The only episodes that are 100% vital for watching series 6 are Silence in the Library and Forest of the Dead from series 4, as they introduce a major character for series 6 (River Song), and The Eleventh Hour, The Time of Angels, Flesh and Stone, Amy's Choice, The Hungry Earth, Cold Blood, The Pandorica Opens, and The Big Bang from series 5, as they continue the River Song storyline, and introduce the new Doctor and companions (which are kept around for series 6).
However, I would still suggest watching all of series 5, as it sets up the relationship between the main companions in series 6, and the entire series has a single plot thread (and it's probably my favourite series of the reboot).
Most of the plot-points for the characters in the first four series are wrapped up by the end of series 4, and David Tennant's run as the Doctor is wrapped up in the special between series 4 and 5. Series 5 is almost a complete reboot, though a few old foes return (or at least make a token appearance).
In the interest of completeness, Dalek, The Long Game, Bad Wolf, and the The Parting of the Ways from series 1 are good to watch to get an understanding of the Daleks and get a basic idea of the plot threads from that series. And Rise of the Cybermen and The Age of Steel from series 2 provide an introduction to the Cybermen.