There are many fine instruments and dulcimer kits available from a number of builders. The internet is a great resource for finding instruments.  However, I have found many benefits to having children build their own instruments.  Originally, I had children build their own instrument from scratch.  Unfortunately, this took a lot of time and required very careful supervision.  This led me to prepare my own dulcimer kits to facilitate the building process.  However, I found several builders who provided very useful dulcimer kits that worked quite well. As a result, over time, I have used a number of different instruments from different builders .  Recently however, I have used the "Cardboard" dulcimer kit available from West Music. I have found it to be a relatively inexpensive instrument that children can successfully build themselves in a fairly short amount of time.  The instrument has a surprisingly pleasant tone and has proved to be a practical and durable instrument that is a good size and weight for children.  The kit comes with cardboard to be used for the body of the instrument plus a wooden fret board.  The instrument comes with a 3 string fret board with the traditional arrangement of frets.  At first, I had my students build the body of the instrument out of wood and used only the fret board from the "Cardboard" dulcimer kit.  However, I had a student who was in a hurry and wanted to use the cardboard that came with the kit.  Everyone liked the sound of her instrument.  And, it was also much easier and faster to build.  So, from then on, I've always had my students build their kits with the provided cardboard and everyone has been very satisfied with the results.   

This sequence of pictures shows the steps for building a "Paper Tune" dulcimer.

This video simply plays the notes for tuning a dulcimer to the key of "C" in Ionian tuning.

Before a child begins to learn to play a musical instrument, they first need to develop an ability to sing.  This develops auditory imagination that enables them to hear melodies in their head so they have a mental reference while learning an instrument.  It helps for a child to already know a melody before attempting to figure it out on a musical instrument. 
This song book provides a sequence of songs that will help a young singer learn to sing in tune.  These songs have been chosen based upon the work of Zoltan Kodaly. They are organized in a sequence that will be particularly useful for the young child as they start to learn to play the dulcimer.  The songs are organized around simple melodic and rhythmic patterns.  These songs are all real songs from the folk tradition.  They have been passed down from generation to generation and are particularly appropriate for young children and the beginning singer.  They have a narrow vocal range and contain only whole steps which are easier for a beginning singer to hear.  The first songs are traditional children chants.  They are songs that children often learn from other children. 
The book also contains simple games and activities that children enjoy doing along with the songs.
Be sure to read:  "Singing with Young Children."
Next read: "Two Note Songs."

This web resource contains demonstrations of how to play the songs on the dulcimer listed in "Song Book for the Beginning Singer."

First, music for children is a social process and it is participative.  Children see other people enjoying music and want to be part of the activity.  The rewards come from the intrinsic benefit of being able to join in and enjoy music with others - not from grades, praise, gold stars, or other rewards.  At first they can join in by listening. If they want, they can clap their hands and dance.  As they hear songs over and over, they pick up the words and start to sing along.  The adults may give them some rhythm instruments such as spoons, shakers, rattles, or a wash board to strum.  They are not being told to do music.  They are given the opportunity and join in through their own interest and volition.  Being able to participate in music becomes their goal rather than someone else's goal for them.  As a result, they begin to enjoy music on their own and they can be seen singing songs and dancing when they are by themselves or with other children.  As time passes, they show interest in the instruments and want to try to play them.  For this a dulcimer is a good introductory instrument.  Adults can show the children how to hold a pick and how to strum.  For this purpose, it is helpful to give children an instrument tuned to an open tuning.  The dulcimer is perfect for this. 

From this point, the child can be shown to strum along with one chord (2 note) songs that use the simple open tuning (doesn't require any fingering of a chord).  Children can strum the beat or the rhythm of the words.  This helps them get comfortable with strumming and helps develop the muscle coordination that will be useful when ready to learn more advanced and complicated strums. 

At this point, it is helpful to bring the child's conscious attention to the idea of a beat versus the idea of the rhythm of the words.  This can be accomplished through a fun activity that invites the children to clap the rhythm of the words of some of the first one chord (2 note) songs.  Then, they can be shown how to clap to the beat of the songs.  When they can do both confidently, they can be challenged to clap the rhythm of the words until the teacher holds up his or her hand.  When the hand goes up, the children are to change to clapping the beat.  When the teacher puts his or her hand down the children are to revert to the rhythm of the words.  Later, they can try this activity as they strum their musical instrument.

While they are still strumming with an open tuning, the next step is to show them a basic rhythmic strum that can be used and practiced with all the first learned songs.  For our purposes, a good strum to use is the "bum diddy" strum.  At first they will need to practice this strum by itself.  When they can play it in a steady and consistent fashion, they can then try the strum along with someone else while the other person plays and sings a song.  As the child get comfortable, they can try singing along with the strum. Children often find it difficult to focus on singing and strumming at the same time.  So, they may need to practice the strum pattern by itself a lot.  They will also find it helpful to strum along while some else also plays the strum and then starts to sing.   

When a child has mastered the ability to strum the "bum diddy" strum while singing simple one chord songs, the child can learn to pick out the melodies for the beginning one chord (2 note) songs. For this, the teacher tells the child to watch carefully where the teacher places his or her fingers on the fret board, and then, the teacher slowly plays the first simplest one chord song (Cuckoo) by playing only the melody notes without a strum.  The teacher can then ask the child if the child would like to see the teacher play the piece again, or whether they child would like to try it.  Since the child can already sing the melody, the child uses his or her ear plus his or her memory to try to pick out the tune.  It is important to let the child try to use his or her ear to pick out the tune rather than the teacher teaching through rote memory.  The teacher can show where the first note is located and reviews where the second not is placed if necessary.  If the child makes the request, the teacher may also replay the piece.  But, it is important to have the child develop an approach of singing the melody and comparing it to what is played rather than have the child try to rote memorize a sequence.  At first, the child is simply figuring out the melody without a strum.  When the child gets the melody, the child may strum the rhythm of the words as they play the melody if they wish.  The teacher should reflect the child's accomplishment back to the child - not through praise, but by pointing out what the child has done - "You figured it out!" 

The child will quickly want to learn some other pieces.  At this point, since the child already knows how to sing all the other introductory one chord songs (listed as 2 note songs), the child can be given another 2 note song to figure out.  Usually, the best song to use is "The Counting Song." The teacher does not demonstrate this song, but instead, leaves it to the child to figure out.  The child may ask if the song starts on the same note.  Of course, the teacher can answer any questions.  The child can be left to figure out the song.  When the child can play the song, it is important to have the child get recognition for his or her accomplishment by being able to play it for the teacher and anyone else.  In addition, it is then fun to play duets.  At first the teacher plays along with the child matching exactly what the child is doing.  When the child shows confidence and  is comfortable, the teacher can say the he or she is going to do something a little different while playing a duet with the child.  As the child plays, the teacher can strum the chord.  The teacher can strum with the "bum diddy" strum at this time.  The teacher can strum on the open chord and then strum on chord one (C chord) in the first position. The teacher can show the child how to play the "C" chord in the first position and then practice the "bum diddy" strum with it.  The teacher can then show the child how to play the melody while using the "bum diddy" strum and let the child try it.  When the child is comfortable using the strum with the melody, the teacher and the child can then play duets and the teacher and child can trade off parts.  Later, the teacher can play a simple harmony part by playing the melody a third lower.  This can then be shown to the child and the term "harmony" can be introduced.  Then, the child can try playing the harmony part.  When the child is comfortable playing the harmony part by him or herself, the child can then try playing the harmony part while the teacher plays the melody.  However, it is important not to introduce too much at one time.  Each step should be taken one at a time giving the child time to get comfortable with the step before moving on to something new.

Now the child can play the melody, harmony, and chord a song with a simple "bum diddy" strum.  The child then adapts what they have learned to all the other 2 note songs by working out all the pieces.  As the child masters pieces, the teacher can point out that the child has figured the pieces out for him or herself, and ask the child how he or she figured it out?  It is helpful to have the child feel a sense of self accomplishment as well as to consciously think about how they have figured out the pieces and what they have learned.

When the child is ready, it is helpful if the child can have an opportunity to play duets and trios with other children.  They can learn to trade off who plays what part.  One child can play the melody while another plays harmony and the third child chords.  In this way, music becomes something they can enjoy with friends and do together.  This can be accomplished right away with the first 2 note songs. 

This represents the pattern and principles for proceeding on to other songs.  This is a developmental process.  It represents engaging a child in pattern recognition rather than rote memorization.  It is learner based more than teacher based.  The teacher provides the tools, resources, model, goals, sequence, opportunity, challenges, and guidance and support,  but the child must actively listen, figure out patterns, and use their understanding to experiment, self check and self correct, practice and review, and use self discipline to work toward perfection.