Definitely get a chromatic tuner, not something called a guitar tuner or dulcimer tuner. There are lots of affordable models in the $10-20 range, but some are better than others. I agree with John that the Snark is one of the better ones in terms of accuracy and ease of use. Because of their low profile, I personally use the D'Addario tuners. One is a headstock tuner that can work on a flat head. But on my scroll-head dulcimers I use the same D'Addario tuner that's for violins or violas. I attach it across the fretboard just left of the nut, where a lot of people keep their capo. If you look at any of my videos when I play the McCafferty dulcimer, you'll see it right there.
If you want to spend a little more, perhaps the most accurate clip-on tuner is the Peterson Robo-Clip. It uses a simplified strobe display that is much easier to read than older strobe tuners. I use that on my guitar.
I also have an app on my phone called G-Strings, which is free and very accurate. The only downside is, like the handheld tuners, it won't work at a jam or whenever there is significant ambient noise.
Dusty T., Northern California
As a musician, you have to keep one foot back in the past and one foot forward into the future.
-- Dizzy Gillespie