Whether goods of two companies using the same trademark fall under separate classes

What is the name of your state (only U.S. law)? NY

I just found another company using my trademark. I started using mine in commerce in 2011. The first I noticed them using the trademark was last month, when I found them on an internet search. Neither I nor they have registered the trademark, and I’m not certain who was using the trademark first. In any case, I’m trying to ascertain whether we would both be permitted to use the trademark by determining whether the goods being sold fall into separate categories. Can anyone confirm whether books / book publishing fall under a separate class from computer software for the purpose of trademark law?

Follow up to the above question

If no one knows the answer about this, can anyone offer any general advice regarding getting information on classes of goods?