While you're down there, remove the trap (the U-shaped piece) and make sure there's not a lot of debris holding stuff up. There's a lot of pipes flowing into that single trap!
Put a bucket underneath, unscrew both sides, dump the water/gunk/etc that's in there. Maybe rinse it out (into the bucket that you keep under the sink or in a different sink; remember, you took the pipe out!). Then screw it back in, add water to fill the trap back up, and check for leaks.
Another posted mentioned Teflon tape, which is often your friend unless there is a compression fitting. These looks like slanted rings that will squeeze against the pipe and seal just fine. These are common in p-traps, but likely not present by the gray hose.