Off hand, assuming you want to use an analogRead to read the piezos for velocity, and assuming you do not have advanced soldering skills, I would say either the Teensy 3.5 or 3.6.
Both the 3.5 and 3.6 have 21 analog pins accessible along the outer edges so that you can use a breadboard to make connections. This way you won't need to solder connections under the board to the other analog pins or add headers for the internal pins.
The 3.5 is tolerant of most of the digital pins being given 5v instead of 3.3v (there are a few analog only pins that must only have 3.3v input). The 3.6 is 3.3v only. But unless you have old devices that emit 5v output, it may not be an issue.
Both the 3.5 and 3.6 have a micro SD card slot that you could use to record things (or read files).
The 3.6 has 11 special pins that support touchRead that allow you to create special analog inputs where you connect the pin to a metal pad and it can read using capacitive sensing. Note, some of these pins overlap other pins that you might want to use for other purposes.
The 3.6 also has support for acting as a USB host. I've not used it, so I can't say how well supported it is.
If you haven't soldered much before, consider getting the Teensys with pins already soldered onto the board. This would allow you to use a standard 1/2 size or full size breadboard to make connections while you are debugging things.
In terms of lights, I would suggest thinking about using WS2812B LEDs (Adafruit calls them Neopixels) or APA102 (Adafruit calls them Dotstars). These LEDs are chained together, so you only need 1 or 2 pins to light up a string of lights, rather than 1 pin per LED. There are some special concerns about WS2812B/APA102 leds, but normally they simplify having many LEDs.
One other thing to think about is rather than using piezos, possibly use a large monitor with touch screen input. PJRC.com sells a 320x240 color monitor with touchscreen support and they have a high speed library (assuming you use certain pins). While you can find the same monitor elsewhere, PJRC actually tests the monitor before sending it out, while some other vendors don't do the test.
Take things slow. Build things slowly, and test as you go. Unless you are extremely lucky, it won't run if you try and assemble everything together all at once, and try it one hour before you need to show off your system.
