Once you have the mechanisms, motors, servos and everything else connected you will need some way to control your creation. There are many options and it really depends on the complexity you require and the money you want to spend. Here are a few different methods and types of animatronic controllers.
RC Remote Control
One of the classic ways to control your animatronic project is with a remote RC style control system. These range from simple speed and steering car controllers to much more complex multi channel systems. The great things with these is that they are already designed to control servos. You can also get motor speed and light controllers that plug in like a servo. The other advantage is that they are designed for battery power which is great if your prop is mobile or located far away from a power source.
Certainly a basic controller you could rig up an Arduino (or any other flavour of microcontroller) and run a few servos, motors and lights with it. The great thing about the Arduino is the huge amount of libraries for various devices (servos, motors, etc) that you want to connect. There are also several cheap MP3 modules available (DF Player, YX5300) which makes adding sound much easier. This will work for fairly simple applications but will soon get difficult if you have many things to control and want to sync the audio to movement. I am not aware of any animatronic specific software frameworks for the Arduino, there are several for robotics that you may be able to make use of including Artoo and GoBot.
For an example I have used a ESP8266 controller, a servo and some WS2812 pixel leds to make this prop gauge: https://vimeo.com/188603358
Octobanger is a DIY Arduino based 8 channel stand alone prop controller project. With it you can synchronize the 8 outputs with an audio track. The outputs are only on/off ( hence the name “banger”) not PWM but still a great controller for the cost. It uses an Arduino variant and a cheap MP3 player to give you a standalone controller. The maker has also written an application that lets you download the sound to an SD card (which goes into the MP3 player) and lets you sync the actions with the sound (this file then gets downloaded to the Arduino). All in all an inexpensive and easy way to get some synchronized movement for your prop. You can get more information on Octobanger (and several other animatronic projects) here: http://buttonbanger.com/?page_id=164
Banshee is a much more powerful and feature laden standalone prop controller. It is PICAXE based and has Servo, PWM and MOSFET outputs along with protected trigger inputs and a MP3 player. You can find more information and purchase the PCB (which includes some hard to find parts) here: http://www.haunthackers.com/banshee/index.shtml
Lightman’s Servo Controller
This a DIY Propeller based stand alone prop controller designed by Brian Lincoln. It has 8 Servo outputs and a MP3 player. It allows you to record the servo movements synchronized with the sound track. You can find much more information and find examples starting here: https://www.diychristmas.org/vb1/showthread.php?2631-Servo-Recorder-Player-and-Servo-Player
Visual Show Automation (VSA) is a windows based application that lets you control a multitude of Servos, Lights, pretty much anything via serial, USB and DMX. It lets you synchronize many channels (128 or 256 depending on version) of pretty much anything with a sound file. You can find more information here: https://www.brookshiresoftware.com/vsa_overview.htm
I have also created a FAQ page on using VSA that you can find here: http://zappedmyself.com/animatronics/vsa-info-faq/
Here are some videos of a few projects. For all of these I have used the BobCat DMX servo controller and VSA: