New bike/fazer owner, proper way to go about the initial service?

[kgjersda]

So I checked and adjusted the chain yesterday. I had to use some rust remover to get it open. In this process I got some of it on the tire along with some other lubricants.
I'm a bit paranoid when it comes to slippery tires, what is the best way to clean a rear wheel tire? I've seen some specialized expensive cleaners in the shop, but what's the cheap method?