New Garbage Disposal, sink won't drain

Hi All, so my kitchen sink is clogged. The water has been draining very slowly for several months. I finally had some time and took apart the p-trap, which was dirty indeed, but not clogged. I attempted to stick a mini-snake into the pipe that goes into the wall and I got it partially in, but couldn't get it through all the say. Likely because of my amateur snake. Anyway, sure, it was dirty just like the p-trap, but couldn't determine if something is clogged. So I put everything back into plan and now the the water won't go down. I thought what I just did would have at least made some kind of a different, for the better. The garbage disposal appears to be running. I thought I'd give this another try before calling a plumber. Could the problem be the disposal even though it runs? Any help would be greatly appreciated. Thanks.

I normally don't do any plumbing buy am willing to try this small job. My bathroom sink drain has rust showing around the edges and my understanding is the only way to get rid of it is to remove the drain; clean the sink area and install new drain. I have included some images, one is blurred but will work. What are the steps for me to do this. The p-trap on mine looks to be glued on the side close to the wall, see image. Cold this be left on and remove from above that? I am assuming the water is fine to leave on to the tap, will put a small container under for the bit in the pipes. Pipes under are plastic so this I am thinking will make it a little easier to work on. Watchedf a few youtube videos but most are for the pop-up drain types and one used white silicone under the flange before installing which seemed a little strange? Thanks for any tips or steps on how to do this. Here is one I found that appears to be fairly easy with the steps shown, mind you I don't have a pop-up stopper. [url=http://www.prettyhandygirl.com/replacing-a-sink-drain/]Replacing a Sink Drain - Pretty Handy Girl[/url]