The best way to tell if your kitchen drain is easily fixable on a double sink with a garbage disposal is to fill the clogged sinks about 3 inches deep, place the basket strainer/stopper on the non-disposal side of the sink and hold it down. Turn on the disposal and if the water goes down on the disposal side, turn it off. If the water does not refill the sink, pull the stopper from the other sink. If that water drains, you just fixed your stoppage.
If the water does refill the disposal side of the sink, the stoppage is past the vent and you need to call Drain Doctor Plumbing and Drain to cable the drain.
If the water on the disposal side just swirls around and does not drain out, turn off the disposal. The stoppage is before the vent and may be in the P-trap under the sink. You can place a bucket under the P-trap and loosen the nuts on the P-trap and remove the trap and see if it is clogged.
WARNING! DO NOT DO ANY OF THIS IF YOU ALREADY USED AN ACID DRAIN OPENER. CALL DRAIN DOCTOR PLUMBING AND DRAIN.
Caution: This process my damage or cause leaks in old chrome plated metal pipes under the sink.