The Teagan T-shirt is designed as a close-fitting tee and is best suited to knit fabrics with some stretch, such as jersey. Cotton knits are often an easy-to-sew option for those new to knits. Rayon knits generally will have a bit more drape and are more slippery to sew.
If you're new to sewing, look for a knit with a few percent spandex for stretch and recovery (snapping back after stretching). Fabric weights can also help you make your choice. T-shirts are typically made in lightweight fabric. A weight of 130-200 grams per square meter (or 4-6 ounces per square yard) is likely to be about right.