I'd try the easy stuff first - make sure your cables move freely and that your derailleur is cleaned and lubed. I think trigger shifters rely on the cable pull provided by the derailleur spring to shift in one direction so anything you can do to increase that pulling force should help.
If things are still gummed up, you can try to re-lube your shifter. Usually, you can take the plastic covers off a shifter without harm but I wouldn't disassemble one any more than that or you'll probably never get it back together. Carb cleaner will degunk it pretty well but is a bit toxic for my tastes. Cleaning the shifter in a dish washer is a safer alternative. To re-lube, I'd use moly grease - the kind used for CV joints in cars. You can get a small amount at most car parts stores for $1 or so. It's pretty thin and shouldn't get too stiff when cold, but be careful handling it - it's very messy stuff.