You aren't *splitting wallets*. Please read how Bitcoin works or watch some videos.
Is there a Bitcoin wallet that supports all three and will allow me to send each coin to a different address?
Not a wallet that can be reasonably recommended.
I have a Bitcoin wallet which is over a year old and contains Bitcoin, Bitcoin Gold and Bitcoin Cash, how do I split these into three separate wallets?
Which wallet are you using? Core, Electrum, Armory, other?