You get new mail and send mail you've already written using the Get Mail command. You can trigger it in any of several ways:
Click Get Mail on the toolbar.
Choose Mailbox→Get All New Mail (or press Shift-⌘-N).
If you have multiple email accounts, you can also use the Mailbox→Get New Mail submenu to pick just one account to check for new mail.
Control-click (or right-click) Mail's Dock icon, and choose Get New Mail from the shortcut menu. (You can use this method from within any program, as long as Mail is already open.)
Wait. Mail comes set to check your email automatically every few minutes. To adjust its timing or turn this feature off, choose Mail→Preferences, click General, and then choose a time interval from the "Check for new mail" pop-up menu.
Now Mail contacts the mail servers listed in the Accounts pane of Mail's preferences, retrieving new messages and downloading any files attached to those messages. It also sends any outgoing messages that couldn't be sent when you wrote them.
The far-left column of the Mail window has a tiny Mail Activity monitor tucked away; click the square icon at the bottom of the Mail window to reveal Mail Activity. If you don't want to give up window real estate, or you prefer to monitor your mail in a separate window, you can do that, too. The Activity Viewer window gives you a Stop button, progress bars, and other useful information. Summon it by choosing Window→Activity Viewer, or by pressing ⌘-0.
Also, if you're having trouble ...