One ways to achieve a stretch in crochet projects such as a bikini bottom is to use a yarn with an elastic blend built in, such as Cascade Fixation Cotton/elastic yarn blend. Another way to achieve stretch and hold is to use a cord elastic in tandem with yarn.
Cord elastic is especially helpful in bulkier projects, such leggings and hats, because the yarn can be a heavier quality. The type of project dictates the type of yarn used. Lightweight projects should be done with lightweight yarns, such as cotton. Heavier projects can be done with heavier yarns, such as acrylic and wool. Both options are widely available and sold both online and in stores.