How to allow/setup crypto transfers/payments on my website?
To check payment:
you can use blockchain api to do so:
https://blockchain.info/q

which is

https://blockchain.info/q/getreceivedbyaddress/[BITCOIN ADDRESS]?confirmations=6

confirmations can be any but minimum 6 is recommended

to accept bitcoin, I suggest u host a bitcoin core and use something like  Easy Bitcoin PHP Library to do so