I found a pattern for this skirt for an adult and fell in love with it. I adapted it to our doll and here it is, the red lace skirt!
You will need a 0.75 mm steel crochet hook (US size 12) and a 100 g cotton tread size 10 or 20 (I used DMC brand). For an American Girl doll I would use a 1 mm steel chochet hook (US size 10) and a cotton thread size 10.
Make a chain to fit your doll's waist (give some allowance to be able to pull in on later as dolls' hips are usually wider). This skirt has about 170 chains. Close it to form a circle. The skirt is seamless, so you will be crochetting it all around.
For the belt: single crochet the first row, then double crochet for 7 rows.
Divide the number of stitches by 4 and start following the chart (you will be repeating it 4 times around the skirt). For the main pattern you will need 31 stitches (times 4). You will need to work out how many stitches should be done in-between. In my case, it was 13 and I took away one stitch in every next row (thus, it took me 13 rows to finish the triangle, and it finishes after the third row of "pineapple" patterns, not after the 4th one as shown on the diagram.).
This project is neither fast nor easy. It does take time to make it but the result is worth it!