Cotton yarn is pretty hard to work with because it doesn't have much give, so I'd go against using it unless you're looking to make a heavier, warmer blanket - however, I am personally a fan of how cotton feels because it gets softer as you wash it and doesn't have a plasticy feel.
Wool is definitely a no-go if you don't like itchy textures.
Acrylic is a good starter and it's textures vary depending on the company and make. It's stretchier, cheaper, lighter, and washes well. I personally dislike the fact that it's made of synthetic materials, but if you don't mind the feel too much than that shouldn't be a major concern.
There are also yarn blends, where you can get a mix of acrylic/cotton/wool or rayon (yarn made from bamboo that adds shine).
Basically you can use just about any yarn. When it comes to knitting or crocheting, you just have to make sure you like the material and have the correct size needles.