I want to buy BTC, take it off the exchange, and just sit on it.
Great
I want to regularly add more to the wallet so an easy transfer method.
If you mean that you receive funds to the exchange and you want to send them to your cold storage then okay.
Should I buy a laptop on it's own, and download nothing but electrum on it to store the BTC?
Well, if you don't have any laptops available and you want the electrum way, then yes. Instead of buying a new laptop it would be pretty easier and more secure for you to buy a hardware wallet, since you stated that you're new to bitcoin.
Would using a VPN be sensible when accessing the internet from said device?
Sensible? How's that related with your wallet?
A hardware wallet is something like Nano or Trezor?
Yes.