Water and drain problems

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When I was washing dishes in the kitchen sink today I noticed that the water would accumulate and once I turned it off it would drain but very slowly. I took a drain snake and put it into the drain to see if it would bring anything up. There was a tiny amount of slimy gunk that came out but that was it. The issue still remains. I wonder if the clog is somewhere deeper. I thought of trying to use a plunger, but the only one I have at home is the one I used to use for the toilet. No way am I putting that into the sink. I found something online that said you can try unclogging the sink by using vinegar, baking soda, and hot water. I wonder if that’s a good idea or a bad one. There was also a suggestion of disconnecting some of the pipes and seeing if the blockage is there, but I’m afraid to do that because I don’t want to damage anything. The last thing I want to do is to call a plumber. Any suggestions on how to try to unclog the kitchen sink? Aside from the clogged kitchen sink, there is no hot water in the house. I was taking a shower in cold water today and some days prior to that. I went to the HVAC room to look at the water heater. Usually there’s a light that blinks. There’s nothing blinking on it today. And the floor next to the water heater seemed to be a little damp though there was no puddle there. I wonder if that means that the pilot light is out or if there is a more serious issue. The last thing I wanna do is to need to replace the water heater. Maybe all that needs to be done is for the pilot light to be relit but I don’t feel comfortable doing that myself. I don’t know what to do.