newbie here...

i recently picked up a 2004 Specialized Allez Elite road bike. Seems to be in pretty good working order except for a worn chain that i'll be replacing soon.

however, when i shift from the smallest chainring to the middle chainring my chain drops about half the time. i'm not shifting under heavy load or on a particularly steep decline or incline. anyone able to educate me on what could be causing this and what the possible fixes could be? will a new chain help at all?

i know i can take it up to my LBS, but i'd like to learn to do much of the maintenance/repair work on my own.

thanks!