Modal or bamboo underwear - soft, stretchy, silky, the solution to everything I hated about cotton underwear. They are usually more expensive than Walmart (or other box store) brands and take more care when washing, but to me are definitely worth it. You can find them for fairly good prices on Amazon if you don't want to splurge on name brands, but you have to do the research and check the reviews. You can get them in boxers, briefs, and thongs for men and even more styles for women.
Bamboo viscose is a very soft and breathable textile.
There are many types of viscose fabric and from what I have gathered the last few weeks the most eco-friendly and breathable seems to be bamboo viscose and lyocell viscose (usually made from eucalyptus).
There's a lot of uncertainty around the eco-friendliness of the production of bamboo viscose, but from I can gather it's a lot better than most cottons and a lot better than polyester (both for the environment and for our skin).
I could go on forever about the different fabrics, but to keep it short if you want soft and breathable underwear go for either bamboo viscose or lyocell viscose.