I have come to the HD with little or no instrumental experience. My formal training has been in voice and choral singing. I was a mediocre violinist starting in the fourth grade; I left that behind to join my High School chorus. I do have the advantage of having learned the notes in both treble and bass clef, and I read music well because I have a good ear (not because I started with the music theory). Any theory I know I have picked up along the way.
I agree with David that there are a number of things you would need to learn to understand the HD, and I think you may be just as well to start out with the instrument. You may actually find that you don't have to unlearn some things where there are differences between the truly chromatic instruments (like the piano) and the diatonic layout of the HD. Jess Dickinson's videos and Christopher Foss's video on "demystifying the hammered dulcimer" spring to mind immediately as great places to start - maybe even before you decide which hammered dulcimer to buy.
I would also encourage you to go ahead and get a package. In the beginning it's much easier to let someone else make decisions about the stand and hammers and tuning. If you have a chance to go somewhere like Song of the Wood and try out different instruments, you might be able to let them help you with a "package." Jo Ann and her staff are a joy to work with, and you can't beat the opportunity to play so many different instruments. If you don't have that chance, I can recommend Chris Foss and his Phoebe package (www.songbirdhd.com
); it was my first HD, and Chris is incredibly easy to work with.
This is a great site, and you won't meet more helpful and wonderful folks than those of us who play the Hammered Dulcimer. There are also a couple of Facebook sites that you might want to ask to join, since there a any number of helpful posts about such things as lessons, workshops, and used instruments on those pages. One is "Got Hammered Dulcimer?" and the other is "Hammered Dulcimer Players."
Good luck and welcome!