Plumbing – Pipes flooding downstairs

If the sink is backing up into the toilet, then the clog is at or after the junction of the two in your drain lines. You'll have to snake the drain to remove the clog, and it will likely be beyond the reach of a standard toilet auger. You could remove the trap on your sink and attempt to snake it from there, or remove the toilet and run the snake down that drain.

If you do remove the toilet, you should wait for the water in the lines to recede so it doesn't flood your floor. If you go through the sink drain, you'll likely have a smaller drain line to maneuver the snake and it's possible the clog is further away, requiring a longer snake.

I'd start by snaking the shower drain. Liquid drain cleaners help with grease and oil buildup in the trap, but less so on hair or other solids, especially if they are further down the line (e.g. near the vent). The shower drain will be a lot easier to snake since you don't have to worry about damaging the toilet's porcelain finish.