The main bathroom sink and tub drain really slow. I thought I corrected the problem a couple of months ago by taking the trap off my sink and using a small snake to unclog the pipes. It worked, or so I thought. Now I have the same problem. It is only these two drains. Do I try the snake again or is there something else I can try? Oh yeah, I have a basement and city sewer. Thank you.

Since it is only these two drains don't bother opening the clean out in the basement, the clog is not after the clean out. Remove the drainfrom the tub and make sure there is no hair getting caught on the plunger. Rod again from the sink after removing the trap. Put at lease 10-15 " of rod into the drain. When you say a small snake how small are you using? A little 1/4" snake may only be pooking a hole in the clog, go larger.

I agree... You may just be pushing the slug further down the system. Work with a larger rotary snake (even if you have to rent one) I would assume the rest of your home drains are OK? Hope this support helps!