there are many ways & styles...some involve chording, and some are mostly playing a melody string with harmony strings. There are also various ways to set up the strings (3 string vs. 4 string) and LOTS of tuning variations.

easiest way to start is very much like what you have been doing...picking out melodies and getting used to the feel of it. Then add a good book like the Jean Ritchie book for traditional styles or a couple others for more 'uptown' stuff...

you also will gravitate toward finger picking or using a something like a plastic pick.....the wonderful think about a dulcimer is that you CAN be playing recognizable tunes pretty fast.

have fun!