The best exchange to use would depend on some factors, like your locality, your trying currency; fiat or some other cryptocurrency and also your stand on identity verification.
You can purchase through binance; they have a limit on withdrawals that can be made without KYC.
Localbitcoins is also a good medium to purchase.
I'm in the US. When you write KYC, are you referring to verification like the Google Authenticator app?
Does it take a few days to deposit a couple thousand dollars or can an of these exchanges take deposits within a few minutes?