Arrgh, it's hard to talk about this stuff without drowning in music theory. Let's take it one key at a time and I'll try not to swim too far into the deep end. Assuming your dulcimer is tuned to the key of D, get your chord chart and learn four chords:
D-Major This is the "root" chord, built on the first note of the D major scale. It is the "one" or I chord (we use Roman numerals because Arabic numerals get used for other things and music theory is confusing enough already).
G-Major This is built on the fourth note of the scale (count up D-E-F#-G on your fingers). It is the "four" or IV chord.
A-Major (or A7) This is built on the fifth note of the scale. It is the "five" or V chord. You can play it as a "five-seven" or V7 if you like the sound better.
B-minor This is built on the sixth note of the scale. It is the "six minor" or vi chord (the Roman numeral is lowercase when the chord is minor). Why is it minor and not major? It has to do with the notes in the D major scale.
With these four chords you can play thousands of folk, country, rock and pop songs. Here's a short list of 3- and 2-chord songs. Adding the vi chord earns you rock progressions like this one. Also this one for aging Boomers like me. Go to chordie.com and find songs to play.
Why are these the chords you need to know? Which notes go into each chord? What's the deal with root (I), subdominant (IV) and dominant (V)? How do you add melody? These are all fascinating topics but they take weeks... months... years to digest. Meanwhile you should be playing music. The more you play, the more you'll start hearing when to change chords and which chord comes next. Until you can do it by ear, find chord sheets online or buy fakebooks. Unlike dulcimer tab which is hard to come by, guitar/uke/mandolin chord sheets are widely available and work perfectly well because a D-Major chord is a D-Major chord no matter what instrument you play it on. You can transpose any song to the key of D and have at it.
Now then. When you're bored with the key of D, here is the Big Secret that makes you a genius:
Everything you just learned about Roman numerals works exactly the same way in each of the 12 keys. Use your fingers to count while reciting the scale. For example, the key of G has a root chord of (guess what?) G-Major and the scale goes G-A-B-C-D-E-F#. Therefore the chords you need are:
I - The root is G-Major
IV - C-Major
V - D-Major (or D7)
vi - E-minor
In the key of C you play the chords C-Major, F-Major, G-Major (or G7) and A-minor.
In the key of A it's A-Major, D-Major, E-Major (or E7) and F#-minor.
If you can count to six, you know the chords for (almost) any song in any key. The only trick is sharps and flats. If the song is in a key I'm not familiar with and I don't have sheet music, then I Google "notes key e-flat" and it gives me the notes I need. Or I make my best guess (is the next chord a B or a B-flat?). If I guess wrong it sounds terrible and I have my answer.
Are the I, IV, V and vi the only chords that might possibly appear in a song? Of course not. For one thing, we skipped the ii, iii and VII(dim) chords! But even those are just the tip of the iceberg. Still, you can often take a more complicated song and strip it down to just the I, IV and V and still have it work because those chords provide the basic structure of modern Western music. Which puts us way out in the deep end.
TL;DR: Learn four chords and sing a bunch of songs. Learn more later.