Step 1 is to find a payment processor. You’ll need to register with a company that provides payment services to small businesses with the payment methods you want. For some, you can sign up with the provider directly (like PayPal), for some you’ll need to sign up with an intermediary (like Stripe for credit cards).
After that you’ll need to integrate it with your website. As for how to do it, it depends on the payment processor, so you should check their integration instructions to learn what’s needed.
That said, you may need to add some server side PHP code to make it work, e.g. to interact with the payment platform to create the payment request. But if you’re going to do e-commerce, you may need some backend code anyways to track and register orders. You could either create that yourself or use an off-the-shelf e-commerce solution to handle it.
In short, there are many payment options, many payment processors and many ways to use them. What works best for you is up to you to decide.