My shower drain is clogged. I have tried everything! When I leave the sink on for very long the drain water comes up the shower drain. Should I put the drain cleaner down the sink drain or the shower drain? The shower drain has water to the top of the drain at this point.

You have a blockage downstream from the shower that must be snaked. This can can be done from one of two places.
1. You can remove the strainer and snake from the shower drain, **OR**
2. You can snake from the lavatory roof vent, Put out enough snake tom reach the base and 15 more feet.
Good luck, Tom

I've tried the hardware store snake but can't get it down very far. I can turn the corner but come up against something about 6 inches further that stops it. When I bring it back up there isn't a hair on it. I keep trying but can't get it past the same block which feels like I've hit a brick wall.

You are in the J bend of the trap. This is a return bend but if you crank and push at the same time you will get past it. Trust me on this. Once past 56the trap you'll only have 8 more feet to put out.
Good luck, Tom

Tom is correct on both counts. You have a clog and may also not be using the right tool. Do you have a drum auger? That should get past the bend. Cobra 83250 1/4-in x 25 Drain Cleaning Tool fairly cheap to buy.