We want to travel from cinque terre in italy to bordeaux. Does anyone know of the easiest way of doing this? Can it be done in one day? Is one way much cheaper than another?

The easiest way would probably be to fly from Genova to Bordeaux but I don't know of any direct flights. Italian no-frills airline MyAir flies to Bordeaux from Bologna (3 1/2 to 4 hours from La Spezia via e.g. Parma or Voghera) and Orio al Serio (Bergamo, one hour by bus from Milan).
Otherwise, if you want to do it by train by day, it would be a long day and the fastest way (13h15) may be via Torino and Paris! Departing Genova (Piazza Principe) at 07:08 arr. Bordeaux at 20:23. The most logical-looking way would be from Genova to Ventimiglia then Nice then Marseille then Bordeaux, but if train times aren't convenient this would only be feasible if you want to stop over at any of these. Therefore you might want to consider going overnight, leaving Genova PP - on a Saturday, at least - at 16:55, taking the night train from Nice Ville (20:58) to Toulouse-Matabiau (arr. 05:14) and departing from there (05:27) for Bordeaux-St-Jean (arr. 07:39).
I would suggest you look up timing for your dates on http://bahn.hafas.de/bin/query.exe/en

You were an unbelievable help. We just got onto myair website and they are having a sale. Picked up tickets for all inclusive 65 euros per person from bologna to bordeaux. Do you think it may be worth our while to hire a car from rome onwards. Or is there public transport from cinque terre to bologna?