You definitely want a powered snake.
We use the general wire ones with a small drum attached to the end of a drill.
When I do kitchen lines that have galv or cast iron fittings in them I usually use the 3/8 cable with a drop head on it. If you can get the cable to go DOWN the drain instead of up the vent, you probably could use a grease cutter head, but it kinda depends on what you've got down there. Could be a hair buildup from the lav even, so a boring gimlet head would be good for that. You might have to run it down several times to fully clear the blockage.