What to do about a toenail abscess at home, such as soaking?

Maybe. If this is a paronychia (small area of pus at edge of nail) soaking may allow it to open and drain. If it persists, gets red outside of that area, becomes increasingly swollen and painful, or have any systemic signs of infection (fever, chills, loss of appetite, etc.) see your doctor asap.
It depends. If this has just occurred then soaking the are in warm water with epsom salt & applying a topical antiseptic cream may help to settle the abscess. If it persists for seveal days it may need to be opened and drained and may need an antibiotic.