If you're unsure what needles you need, this is a great starter pack, giving you a variety to experiment with! 50 assorted hand sewing needles with a needle threader.
The different types that come in various sizes are:
Tapestry needle: Short with a blunt tip for cross-stitch and similar (often good for wool underwear too)
Chenille needle: Short with a sharp tip for tufting, but also useful for some embroidery with thicker thread as it has a large eye that is easy to thread
Darning needles: For darning!
Sewing needles: For seams typically done on a sewing machine, like closing torn seams or other tasks with thin sewing thread.
Quilting needles: Shorter than sewing needles for quilting, tailoring, and similar (faster to sew with if you're making short, individual stitches because the needle is short).
Embroidery needle: Best suited for embroidery, but the eye might be a bit small for un-separated stranded cotton. However, it's thin, sharp, and versatile.