Would I be violating trademark infringement laws if I use the name "Chicago" with a phrase on shirts

Tom,
The word Chicago is in the public domain in terms of copyright so you are free to use it. However, in terms of trademark infringment the matter may be a bit more complex. Without having seen your design, It is likely that you are probably not infringing on another mark so long as you are not using a design which is similar to other marks and designs already in commerce, such as the logo for the band Chicago or teams like the Cubs or White Sox. I would be happy to review your design and discuss intellectual property protection with you further.

This response is informational only and is not legal advice. Further, this response does not establish an attorney/ client relationship.

Andrew's answer is spot on.

The word "Chicago" is free for all to use -- except when used either alone or within a phrase or logo in such a way that it infringes someone's trademark rights. Without knowing what phrase that you intend to use, no one can tell you if you'd be infringing someone's trademark rights.

The word Chicago, all by itself and within a logo, is a federally registered trademark for a number of companies. See:
http://tarr.uspto.gov/servlet/tarr?regser=serial&entry=78326527
http://tarr.uspto.gov/servlet/tarr?regser=serial&entry=78603614
http://tarr.uspto.gov/servlet/tarr?regser=serial&entry=78324342

In addition, there are many hundreds (if not thousands) of phrases that contain the word "Chicago" that are used to brand goods and services (that is, used as trademarks) and many of those phrases are federally registered trademarks.

In short, before you sell clothes emblazoned with any phrase, you need to ensure that the phrase (or any confusingly similar phrase) is not already being used by someone as a trademark.

As you've been advised, the city name Chicago is not protected (and city and other governments can't, for the most part, even own copyrights), but if you create a stylized logo like the rock band Chicago's, I bet that's trademarked for entertainment services and for clothing.

Short phrases are also not protected, such as "I (heart) Chicago."

You're best off consulting with an IP lawyer to evaluate the exact uses you have in mind.

Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.