When we bought our first home the older woman across the street (Interrupting Ida for her stellar timing on phone calls to a newly married couple, ROFL!) advised putting 1 cup baking soda and 1 cup vinegar down each drain once a month. It handles grease, is easy on the environment, and is CHEAP.
Lately there is a thingy in the plumbing section of Wallie World. It is a long white plastic thing you stick down the drain. It catches the hair/junk and pulls it out. It is a one use thing and was $2.26. AWESOME for a house with two females with thick long hair.
Never thought about oxiclean and club soda. I may try that. Hmmmm......
It is a good idea to have a plumbing snake on hand - the thing you put down the drain that turns? A very wise investment. We did the white plastic jobbies because we have some strange tubs with not enough room to get the snake in. I have to mess with them more than my hands will do sometimes to get the snake in.