However, it's cheaper by train. (By the time you've allowed for airport check-in times, it might also be quicker). Unfortunately, the National Rail website shows that there are no discounted fare bargains (of the type where you buy two single tickets instead of a return) available for this weekend. You'd have to pay �91.40 for a Saver Return. (When you get to London, buy a One Day Travelcard for �4.90. Single tube fares, in central London, are �3 per journey, so you will save money with a Travelcard).
Discover Dogs is at Earls Court 2, which is by West Brompton tube station. If you opted to fly to Stansted, your train from the airport would take you Liverpool Street. Take the Central Line, westbound, for 11 stops to Notting Hill Gate. Then take a southbound District Line train (showing 'Wimbledon' as the destination), for 3 stops, to West Brompton.
If, as seems more likely, you opt to travel by train, you will arrive at Kings Cross. Take the Piccadilly line, for 11 stops, to Earls Court. (The signs will either say 'southbound' or 'westbound'). You can walk from Earls Court or, alternatively, transfer to a southbound District line train (with the destination shown as 'Wimbledon'), for 1 stop, to West Brompton.
There's a map of the area here:
http://www.thekennelclub.org.uk/download/249/e arlscourtmap.pdf
You can find out more about Discover Dogs, including the events schedule, here:
http://www.thekennelclub.org.uk/buying/discove r_dogs.html
Chris