Under the sink you should have a curve that goes down from the drain, then comes back up, then into the wall. You should be able to unscrew two places and take the “macaroni” shaped part out, and clean out the whole vertical part.

First shut the water off. This is just in case you bump the faucet levers, not strictly necessary, but a good safety net for a mess.

Second, put a shallow bowl underneath, as some water will come out.

Third, unscrews the two nuts that hold the lower curved piece in place. These parts will stay on the wall side (just slide them up and out of the way), and will come off the vertical piece. As you do this; note where the sealing rings/washers are and which direction they face.

Fourth, if your sink has a drain plug you control from above, you may need to unscrew the small control arm under the since that connects to the drain plug assembly. It’s a steel rod with a small plastic ball on it. Remove that part first, and then lift the plug out from the sink.

Fifth, at this point you should have a clear, open tube between the sink and your bowl underneath. Taking whatever you have to push and scrub to clean it out.

Sixth, put it back together in reverse order. Some Teflon tape on the drain control “arm” where it screws into the faucet would be wose