Looking for some free or low cost activities? Here's some that I have come across in the past month.

Providence Children's Museum: http://www.childrenmuseum.org/htdocs/index.html
The first Sunday of each month is free, thanks to CVS. And for the month of June '06, each Friday evening 5:00-8:00 P.M. is free, thanks to MetLife.

Audubon Society of Rhode Island in Bristol: http://www.asri.org/
The first Saturday of June and July '06 are free, thanks to Citizens Bank.

Roger Williams Park Zoo: http://www.rogerwilliamsparkzoo.org/
If you live in Providence (excludes East and North Providence), the first Saturday of every month is free with proof of residency, thanks to the College Boundfund.
June 18, 2006 is "Day for Dads" - all dads, grandads, stepdads, and honorary dads get in at half price when you bring a child with you.
New Year's Day is free to everyone.