Two primary causes of drain "clogs": A physical clog, like a huge hairball blocking the pipe, OR a venting problem... which mimics a physical clog.
For a physical clog just snake the drain or use pressurized air.
For a drain to work properly it MUST allow air in, or you get the effect of a straw holding water. Either you have a vent pipe connected to the drain pipe which extends vertical through the roof, or you have an air admittance valve. Air admittance valves can stick closed not allowing air in. A "burping" drain is an obvious symptom of venting issues.
Many times the shower is directly above the gray tank and the plumbing doesn't require a vent... the run is so short that the tank vent is all it needs. However, sinks nearly always need a vent... look under your cabinets for an online air admittance valve and see if it's stuck closed.