i sympathise - i am allergic to many animal yarns and am a vegetarian so don't use silk. frankly from what i've heard about the treatment of sheep i would think twice about buying wool even if i could wear it.
i think the best non-animal yarns for most purposes are acrylic-cotton blends, eg rowan all-seasons cotton. they have more elasticity and are less drying on the hands than pure cotton, and are more breathable than acrylic, which doesn't absorb moisture, so if you sweat while wearing it, you stay sweaty.
your options are really limited to synthetics like rayon and acrylic, cotton, and the lesser available hemp and linen, but you can get a pretty wide variety of interesting yarns from these.
i am not well-informed about the dyes used by yarn manufacturers but you may want to investigate the possibility that anything red, brown, purple, pink or orange may have been dyed with cochineal.
ann points out that synthetic fibres use chemicals and processes which may be harmful, but i must point out that any fibre is processed - i mean, wool does not grow on the sheep in ready-coloured 4ply strands which are cut from the animal by nomadic shepherds with flint hand-axes, and then transported to the shops in horse-drawn carts! i mean, unless you live a seriously alternative lifestyle it's a bit much to expect that nothing you do will harm anything at all.