If you don’t mind ordering online, you might want to check out Dharma Trading Co’s line of natural fiber dyes called Procion. They’re one of the more intense, color-fast dyes I’ve encountered for cotton, and they have kits! They also sell many different dye-able garments including t-shirts stitched with cotton thread so that the stitching dyes too instead of staying white (most commercial t-shirts are sewn with polyester thread which does not take). http://www.dharmatrading.com/
Aljo also has really good dyes with similar color-fastness, but they’re more geared towards professionals. http://www.aljodye.com/main.html You can totally order and use their dye as a hobbyist, but dharma is just generally more user-friendly if you’re not a serious dyer.
Finally, if you just wanted something readily available, most grocery/drug stores carry Rit or Tintex in the laundry aisle. It’s very easy to use, develops with salt as opposed to something harder to find like soda ash, and gives good color.
Whatever you use, be sure to wear a particle mask when handling dry dye powder and protective gloves when handling dye in any state. This goes for soda ash as well which is rather caustic. Have fun!