My step dad is a plumber. His advice -- NEVER use draino! (I have never met a plumber who liked the stuff... it's caustic and it is a hazard of their profession).
Buy yourself a plumber's snake. (A wire hanger is an acceptable DIY substitute, but the snakes are better if you have tons of hair. And actually, since a plumber's snake doesn't cost that much, it is a worthwhile investment for families with lots of hair!)
If you have old drains, you may also need to do the baking soda/ vinegar trick after you unclog them, since there will still be grease stuck in there (yuck!)
If after that, you still have problems, call a plumber. (They have professional electric snakes that go really deep into the pipes). I can guarantee you the plumber will bless you for not using draino!
HTH!