All great responses thus far. I've lived in Saint Louis for the past 4 years. We have boys just a bit younger than yours (3 year old twins). Here's my take on a few things, with extra hints to help things go more smoothly
Forest Park (FP): The largest park within city limits in the US. About 2.5x the size of Central Park in NYC. Main Attractions within FP: Saint Louis Zoo, The Muny, History Museum, Art Museum, and Science Center (part of this is in FP, the other part is right across the street). Here's a couple of items about each of these.
Saint Louis Zoo: Largest free zoo in the world I believe. Admission is free, but if you get there late the only parking left will be in the lots, which is $15. Can pack a lunch and bring it with you (or just go back out the car to eat). The best parking that's free is by the North Entrance (lots of 'street' parking within FP just north of the zoo, as well as the small surface lot by the art museum). Get there early to get free parking, and enjoy many of the paid exhibits for free before 9:00 (another poster mentioned this). Children's Zoo (basically a petting zoo) and Stringray Cove are free up until an hour after the zoo opens.
The Muny: One of the largest outdoor theatres in the US. Fairly central within FP, lots of ways to get in/out Broadway type of plays each evening in the summer. Typically a show plays for about a week (so while you're here, there's likely only one show that will play). The free seats area is approximately 1000-1500 seats that are in the back of the theatre. You have to get there typically 1-2 hours before the show to free seats. Depending on how your kids will be in the hot sun from 6-8 PM (show starts at 8:15), I might spring for the $9 seats in the back of the theatre (the rows right in front of the free seats). Still a fantastic experience. Probably your 7 year old would enjoy more, don't know about the 5 year old.
History and Art Museum: Free
Science Center: They advertise that they're one of the top 10 science centers in the US. Free admission, but parking is $10 if you park in their lot on the outside of FP by their main entrance. 80% of the Science Center is south of the interstate, but is connected to the remaining 20% (the planetarium) that is in FP, connected by a cool walkway over the interstate. Park on the FP side by the planetarium, because parking is free, and you can walk to the main part of the science center through the walkway. Omnimax (bigger than IMAX) is really cool for that age, especially if its like an animals/robot/space type of show but costs a couple of $ per person.
Other stuff thats great for STL: City Museum, STL Cardinals game, Botantical Gardens, Magic House (children's museum).
City Museum: Downtown. Admission is $12 last time I checked per person($10 for the Friday and Saturday evening sessions after 5 PM). I am not yet brave enough to take our 3 year olds there. Its basically a massive indoor/outdoor climbing/slide experience. Never seen anything like it anywhere else in the world. Truly have to to experience it. One of their main advertising draws is the 10 story slide (not a typo), but there's much else there that's cooler. Your age range would LOVE LOVE LOVE this probably.....but are you and the spouse brave enough :) I'm not yet. There's crawling involved, I would wear athletic pants (unless its crazy hot). It's right downtown. You can pay $5 to park in one of the lots close to there, or find street parking thats probably free. Google some of the 'city museum' images to get an idea.
STL Cardinals game: Downtown. Sounds pricy, but can be very mustachian. If you go to a game Mon-Thurs, you could theoretically enter in the 'aaa16' code on the Cardinals website if you were a AAA member (or so could anyone else, wink wink nod nod), to get $10 off each ticket. The tickets won't go below $10 with this code (example: an $11 ticket does not go to $1, only goes to $10, but a $20 ticket goes to $10). The crazy thing about the Cards games....you can bring in all the food and drink you want. Soft coolers, and closed plastic bottles are allowed (water, soft drinks....they just don't want people sneaking in alcohol in open containers). My wife and I bring sandwiches and drinks with us, and pay NOTHING for food while we're there. A family of 4 could go for $40, plus the cost of their food they bring in, and sit in decent seats. I've seen people with families walk in with pizza's and paper plates for the kids. Parking is $6 if you park about 5 blocks north of Busch Stadium in those parking garages off Pine Street. Much cheaper, and easier in/out. Also, if kids get bored, there's small playground/batting cages/arcade area close to the Ford Plaza near the outfield seats that you can go to for a bit.
Botanical Gardens: One of the best if not the best gardens in the US. About 2 miles southeast of FP. Admission is not free during regular hours BUT, if you go on any summer Wed evening after 5 PM, admission is FREE, you can enjoy the gardens for a couple of hours, then enjoy a free outdoor concert. People bring food in coolers and lawn chairs.
http://www.missouribotanicalgarden.org/things-to-do/events/signature-events/whitaker-music-festival.aspxMagic House: In Kirkwood. About 25 minutes southwest of downtown. Fantastic children's museum with loads of stuff to do. My favorite area is the detective area where they can solve a mystery (find hidden room behind a fireplace, very cool) Can get very busy, especially if its raining that day. Admission is about $10/person. Free parking.
Good luck. Saint Louis is probably one of the best cities in the world for free/cheap stuff for kids. Have fun!