Roger WIlliams Zoo is fun for all ages! Nice zoo, you can get discounted tickets through some public libraries.
I believe you have to be a RI resident to get a library card and get the discount...unless you know someone in the area who can get the pass for you.
April can be tough...could be nice and warm, but could be quite cool still.
I would recommend Newport, even if you don't visit any Mansions. However, if you do decide to visit the mansions, they it can be quite expensive. Package deals are available if you want to visit more than one. You might also consider purchasing a family membership ($90) which allows access to the mansions with front of the line privileges, 10% discount (I think) in the gift shops and also access to the behind the scenes tour at The Elms, which is very cool. I think for most of the month of April there are only 5 or 6 mansions open though, and some of the smaller ones which won't probably be open then are really cool. Anyway, here's the link to the Newport Preservation Society:
http://www.newportmansions.org/index.cfm
Along Bellvue Ave (where the mansions are) there is also Belcourt Castle that has tours. Not sure if they are open in April. It is not owned by the Preservation Society though, so it is not in their packages. They also do Ghost Tours, as it is said the mansion is haunted.
http://www.belcourtcastle.com/
If you go to Newport, definitely take a ride down Ocean Drive and stop of at Bretton Point to watch people flying kites.
One of my very favorite places in the state is Beavertail Park in Jamestown, the next island over from Newport. It is free to get in, and if it is a nice day, it is the perfect place for a picnic. Beautiful place to go walking out on the rocks and watch the waves. Just don't go wandering around on any of the overgrown paths...lots of deer, and lots of ticks. Stick to the low grass and the rocks. Absolutely beautiful!!
Galillee/Pt. Judith can be fun to visit too, especially of you like seafood. Stop off at Aunt Carrie's for some "chowda" and clamcakes or some "stuffies". If you live anywhere near the water in Maine, then nothing new to you. You could also stop by Iggys for douhboys and clamcakes. We have to do this at least once per summer.
Providence can be nice to visit. the Rhode Island School Of Design museum is nice.
You might want to check out a show at Trinity Rep, which has an excellent professional acting company. They will be opening the Odd Couple on April 9th. You can purchase last minute tickets for $15 if the show isn't sold out if you go to the box office 2 hours before showtime. Also, they offer $10 bench seats (top row of the theater). The theater is small, so you can see fine from anywhere in the theater with the occasional exception of to the far right or left.
http://www.trinityrep.com/box_office/
And there are some really awesome restaurants in the city. Check out Rrestaurant.com and purchase a gift certificate ahead of time, or eat somewhere on Federal Hill for some great Italian. You could also head to the Povidence Place Mall to grab something quick to eat, or have a sit down meal (Cheesecake Factory, the Melting Pot, Fire and Ice) or Dave and Busters would be a fun place for the kids, and the food is good, but it gets expensive once you start purchasing the gamecards!!
Also the Providence Place Mall has an I-Max Theater which can be fun...but better to purchase tickets ahead of time, as they often sell out on popular movies.
http://www.imax.com/
While you are visiting, make sure to get some Dels lemonade!!
Hope some of this is helpful! Enjoy your visit!