What could be causing my new GFCI receptacle to keep tripping?
The washer and dryer are on the same circuit? It must be a gas dryer?

I'd think that you would want to see about splitting this out on its own circuit if you can. The drum motors in the washer or dryer could be the culprit. GFCI only requires a 0.005A difference in current between the hot and neutral legs to trip.

Your GFCI outlet might be defective or over-sensitive.

There could be a small/intermittent short to ground in your outdoor spotlight fixture? Maybe the lamp itself is defective and causing enough fluctuation to trip the GFCI.

Do you have a moisture problem in the outdoor spotlight, perhaps?