At that time of year, the zoo will open at 9 a.m. The carousel and the children's zoo will be free for the first hour. I don't believe the train is also free the first hour, unless they have changed it. The cost for the train is $5/person. Parking for the zoo is costly, but try to find a place on the street. Sometimes it is closer than the lot and it is safe. We have also parked in the Art Museum lot and walked to the zoo entrance. (and then stopped in the Art Museum to look at the mummies and Monets).
The Science Center is probably a little old for your daughter, but it is also free and near the zoo so you can always stop in a look. Park at the Planitarium side in Forest Park to get free parking, not the main entrance on Oakland Ave. What is age appropriate for a three year old at the Science Center is a play room called the Discovery Room. It costs about $3 per person and you stay in the room for an hour. It is very fun and hands on. You will need to buy tickets for it at the ticket counter.
Another place I love for all ages is the Missouri Botanical Garden. My children love to go there--at any age. I started going with my son when he was about one. They do have a children's area (I don't think it opens until April though), but the whole garden is just a delight. Bring your stroller there, though.
The Transportation Museum is great too and not too far away from the Magic House. They also have a hands-on play room for a nominal fee which would be great for a 3 year old. My boys were train fanatics so we were there alot when they were your child's age.
I don't believe that Grant's Farm opens until sometime in April.
Also, check your dates when you go to the Magic House. It is usually closed on Mondays, but will be open everyday during the spring break period. It is also free the third Friday evening of the month.
We also like to go to Powder Valley in Kirkwood, which is a conservation area operated by the Missouri Dept of Conservation. They have a small, but well done conservation museum and paved walking trails through the woods. It is also free. (this is the budget board

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PM me with your address and I'll send you coupons for the Magic House (never pay full price there--there are always discounts). How many people will be with you?