Clearing a clog is really easy. You can try to physically remove it with a plunger or the build up of CO2 with the vinegar and baking soda or you can remove it chemically.
Try the baking soda and vinegar trick but make sure you block any overflow hole the sink might have, or the pressure from the CO2 will just escape out that way. The pressure should push the muck down and out. I have to use Sodium Hydroxide in my bath/shower configuration because of how it is built, but that’s very easy as well, but you must use caution as it is highly caustic. I stand in the tub with rubber boots on as I carefully pour the chemical down the drain, but I’m used to dealing with chemicals. You should be capable of using an over the counter drain cleaner from the supermarket and following the instructions on the back. I wouldn’t go taking pipes apart. Tools and leaks needing plumbers tape or horse hair…. it becomes a whole big job.
Try the vinegar and baking soda. Block the overflow drain. Let the CO2 press the plug out. If that fails, get a bottle of ‘Draino’ or what ever the local brand is at the grocery store and follow the directions on the bottle.
Never never never stick food or fat or coffee grounds down the sink. If you don’t have an easy way to throw it in your garbage, flushing it is a better option. Collect fat in an old tin with a lid. When it’s full, throw it in the garbage. If you can wait for the fat to harden in the pan, wipe it with paper towels and throw it in the garbage. If you need to clean out your fridge, lay out several layers of newspaper on the counter. Dump the horrible contents of your left over containers onto the newspaper and wrap it up and throw it in the garbage outside.