Sorry for the dumb question, but I have very very limited plumbing experience, and it's all getting the water to the house, not getting it out.
My house was built in 1975 and the sink is a side by side with garbage disposal. Lately it has been backing up, and when I hit the garbage disposal it fills up the opposite side. It is also very slow to drain. I took a quick peek under the sink and the drains don't look immediately connected. I tried drain-o down both drains with little to no improvement.

Where should I start?