Don't Catholics Worship Mary?

A reader asks:

Before I re-read volume 1 of “Mary, Mother of the Son” I wanted to ask if you think it is worthwhile or advisable to try to address my friend’s claim that Catholics worship Mary.  For brief background, he is a former Catholic who was poorly catechized and now belongs to a large non-denominational church.

I believe you’ve said that even after your conversion, you aren’t drawn to a strong devotion of Mary, even though you accept all the teachings of the Church (perhaps I am mixing you up with someone else; my apologies if I am).  Since Marian devotion is such a contentious issue for so many Protestants, is my effort better directed at other areas of apologetics, like the Real Presence, authority of the Church, etc., and then working on the Mary angle?

I look forward to hearing your advice if you have any to offer.