There are a couple ways you could go:
Wireless. Charge while playing.
Wired, but still pretty simple.
Yamaha UD-BT01 Wireless USB-MIDI adapter
This adapter still needs to be connected to power from a wall socket or a USB port, but that doesn't have to be from your iPad. You can find it for about $49.
Check under Settings → Bluetooth Settings inside the Synthesia app to pair your keyboard with your iPad. After that it should appear in the music devices list automatically.
You may already have this cable if you have a printer or scanner. The standard A-to-B USB 2.0 cable is easy to find and is only a few dollars.
Do not buy a "USB 3.0" cable. It's a different shape that won't work.
Do not buy expensive $20+ USB cables. There is no reason to pay more than $5.
Troubleshooting If your keyboard isn't showing up in Synthesia, check your keyboard manufacturer's website to see if a driver is available. The iPad doesn't allow the installation of drivers, so this might be bad news. If the keyboard also has MIDI ports, try that instead of USB.
Once everything is connected, your keyboard should show up in Synthesia.
Head over to the Musical Devices section on the Settings screen to enable your device for input and output.
Then on the Music Input screen for your keyboard, play the lowest and highest notes so Synthesia knows how many keys your keyboard has.
Lightning to USB Camera Adapter
The "camera" part of the name is a little confusing, but this lets you plug USB accessories (like cameras or keyboards) into your iPad. You can get it from the Apple Store for about $29.
Do not buy a female-to-female USB adapter for the regular iPad USB cable. It won't work.
You may already have this cable if you have a printer or scanner. The standard A-to-B USB 2.0 cable is easy to find and is only a few dollars.
Do not buy a "USB 3.0" cable. It's a different shape that won't work.
Do not buy expensive $20+ USB cables. There is no reason to pay more than $5.
Troubleshooting If your keyboard isn't showing up in Synthesia, check your keyboard manufacturer's website to see if a driver is available. The iPad doesn't allow the installation of drivers, so this might be bad news. If the keyboard also has MIDI ports, try that instead of USB.
Once everything is connected, your keyboard should show up in Synthesia.
Head over to the Musical Devices section on the Settings screen to enable your device for input and output.
Then on the Music Input screen for your keyboard, play the lowest and highest notes so Synthesia knows how many keys your keyboard has.