I give a 6 year old guitar lessons, here is some of my experience.
1. Usually they will want to pay one song over and over
2. She has a full size guitar and it is nearly impossible to make full chords - like C, although she can do a G, so when first learning had her make part
formation of chords, but now she does full formations when possible.
3. She started with a Hanna Montana guitar - pink
4. Switched to light strings and tuned down a step so she can press on strings and not hurt fingers
5. She picks the songs
6. Children at this age loose interest quickly so play games like try to stump the teacher - see if I can guess the chord(s) by closing my eyes as she plays different chords or what verse is she playing now.
7. Likes to play singing notes which helps with learning notes and chords.
8. She knows the name and position of many chords
9. Like fortyearspickn said about spending on the first guitar, but I will add to make sure the action is a low as possible, if possible to adjust.
10. I am kind of preferential to Yamaha guitars for someone learning as they tend to be well built and sound good for the money.