Ah! My home park. It's very possible to make Animal Kingdom a full-day park, but if you just want to hit the highlights and move to EPCOT, here's what I recommend:
1. Get there early. Grab your Expedition Everest Fastpasses and head right to Kilimanjaro Safaris.
2. Safaris should be your first attraction target. The majority of the animals are out when it's cooler outside (morning), and the line is much shorter.
3. After hopefully seeing all the animals out there (I've had some pretty awesome morning safaris), head back towards Asia for Everest. If you're early for your Fastpasses, hit the Maharaja Jungle Trek, right next to Kali River Rapids. The tigers should also be out in the morning.
4. Make sure you check your times guide for showtimes for both Nemo and Lion King. If time allows for only one of the two, go see the Lion King. Nemo really isn't all that good, and the Lion King musical has been thought to rival Broadway. It is also currently the highest rated show on property as ranked by park guests.
5. MAKE SURE YOU RIDE DINOSAUR. The ride gets a lot of hate on the boards here, but it's an indoor attraction that'll allow you to help cool off for a bit. Scary for the little ones, but it's a really awesome ride. Oh, and if Ryan's working, don't forget to say hi!
6. For lunch: Rainforest Cafe is well overpriced and generally outside of the price range you specified. I'd recommend either trying to make a reservation at Yak & Yeti, or, depending on where you are in the park, trying out one of the quick service restaurants. Pizzafari is by far the best deal in the park. $10 should get you a personal pizza and a side salad. Yak & Yeti and Rainforest Cafe are the only table service restaurants in Animal Kingdom.
7. You should be all done! If time allows, It's Tough To Be A Bug is another nice cool-down, air conditioned attraction. I personally don't like water rides so I stay away from Kali River Rapids, plus the line for it generally exceeds the line for Everest during hot summer days.
In terms of Fastpasses:
If you follow this model, you should only really need Fastpasses for Expedition Everest. Dinosaur tends to fill up on hot days as well, and grabbing a Fastpass there before heading to lunch and a show can be recommended, but not always necessary.
Have fun in the kingdom!