Pere David was certainly the first Westerner to discover the giant panda but he never saw one alive. The first specimen he saw was a skin in a local hunter's house on the 11 March 1869. The hunters promised to obtain a specimen for him and set off the next day, returning ten days later with a young cub which they had caught alive but killed in order to transport it easier. An adult female panda was brought to him several days later, also dead. Coincidentally just two days later he was brought his first (also dead) red panda specimen (which was already known to Western science).
The first Westerner to see a live giant panda was, as Vulpes says, in 1913 (I've also seen dates of 1915 and 1916 so I'm not sure which is correct) , when German zoologist Hugo Weigold was given a live cub by some local hunters. It died shortly after as it had not yet been weaned, and the skin ended up in the Berlin Museum along with the skins of five other adults provided by the local hunters.
The first Westerner to see a giant panda in the wild was possibly J. Houston Edgar, an explorer and missionary who wrote in 1924 that he had in 1916 seen "an animal asleep in the forks of a high oak tree which has puzzled me ever since. It was very large, seemed quite white, and was curled up in a great ball very much after the manner of cats. It was unknown, and a source of wonder, to my Tibetans." Other claimants are Lieutenant J.W. Brooke and General G.E. Pereira. There is no actual proof for any of the three.
The first Westerners to kill a giant panda was a double-team of Kermit and Theodore Roosevelt (sons of President Theodore Roosevelt) who both shot the same panda at the same time on 23 April 1928.
