There are two commonly used methods for paying bills in QuickBooks. One is to simply enter the bill as a check, just as you would with your manual checkbook. The other is to enter the bill and then separately pay the bill when it comes due. This blog deals with the former. For a comparison of the two methods, see Entering Bills vs. Writing Checks.
To enter payments made via check or cash directly into QuickBooks, open the Write Checks window from the Banking menu.
Enter the check details (check #, date, payee, amount) just as you would in your manual checkbook. If you print your checks, then fill out the account or invoice number on the Memo line so it will show up on your check. Then assign the payment to an account and if you track items by customer:job and/or class, fill these fields out. And you're all done.
All done, that is, unless you print your checks from QuickBooks, in which case you still have to print your checks.