Oooh, well, it's a nice little town, only small, so a day's just right for having a potter around. What I like about it is the fact that it's a working harbour so it's a very functional town, not at all twee.
First of all I'd go and see the harbour:
http://www.seewhitstable.com/Whitstable_Harbour.html
I'd have a drink in the Old Neptune on the beach:
http://www.seewhitstable.com/Wheelers-Whitstable-Restaurant.html
The beaches are just lovely. Sort of bleak, but in a very beautiful way:
http://www.seewhitstable.com/whitstable-beaches.html
Harbour Street's got some interesting shops and cafes:
http://www.seewhitstable.com/Harbour-Street.html
If your budget can stretch to it I would recommend having a seafood meal at Wheeler's Oyster Bar. You would need to book - they only have about 4 tables in the tiny restaurant! It's byo booze so that makes the price more reasonable:
http://www.seewhitstable.com/Wheelers-Whitstable-Restaurant.html