A big distinction between amateurs and professionals is that professionals get paid for what they do. The first thing you'll need is a bank account for your business. I recommend you get one that's separate from your personal account. That makes bookkeeping easier and keeps you from having to give personal banking information to clients.
I found the simplest way to receive payments in the United States is to simply ask clients to send you a paper check, then deposit that in the normal way. However, you should also be prepared for electronic payments, particularly if you have clients outside your home country. Beyond traditional banks, you can also get paid through online payment systems such as PayPal. I found it to be great for smaller clients, but because it's not a real bank I do recommend that you sweep money from your PayPal account into your business bank account as soon as the money becomes available.
Although PayPal gives people a way to pay you with credit cards it's not as robust or as cheap as Merchant Card Systems. Those let you accept credit cards directly without PayPal as a mediator. Again, your bank can probably offer you such services or can refer you to somebody who can.
Whatever payment option you use, I recommend that you write it into your contract. For example, I might say, "Payment by check drawn on a United States Bank must be received by such and such date, payable to Bobby Parker" But it's also wise to ask the client how they prefer to pay and then make sure it's possible. After all, the more ways they can pay, the faster you'll get paid.