You could just attend a "mass" there. No charge, and you can just respectfully leave early if you do not want to stay for the entire mass. No need to "dresss up," just dress respectfully...men long pants, ladies no shorts or clothing that is too revealing, etc......leave the ballcaps at your hotel or put in your day bag. After the mass, the lights are then typically dimmed, but during mass it is bright and beautiful. Mass times are listed on the web site. It has been several years, but seems we just entered by the door on the left (which, of course, could have changed by now). Just shuffle in with (or after) everyone else.....do not make a big deal or explain about being a tourist, just "go with the flow."
Not Catholic or never been to a mass before?, just "do what everyone else does" standing, sitting, kneeling, etc. Best to not go forward for communion (and not everyone would), though, unless you would otherwise do that at home. When the collection plate is passed, toss in what you deem appropriate (fair to consider a portion of what would have been your admission otherwise)..
It is a beautiful church. After the service, upon polite request, the ushers let us just go on up the stairs in the back (there is a small fee for that part) to see the upper part (and you can then see a view down into the basilica, see bronze horses, etc, and also a wonderful view outside looking over. Marks Square).