Android offers options for using both Gmail accounts and non-Google email accounts (POP, IMAP, or Exchange). We'll begin with the former.
When you first power up an Android phone, the initial setup process will help you configure a primary Gmail account to be associated with the device. That account will automatically appear in the integrated Gmail app, and you'll always see mail appear there instantaneously as it arrives.
On phones running Android 2.0 or higher, you can add other synced Gmail accounts, as well. While in the Gmail app, press the phone's Menu key, and then select Accounts. Touch the Add account button at the bottom of the screen, and follow the prompts. Once the setup is complete, you can toggle among your various Gmail accounts by pressing the Menu key while in the Gmail app, and selecting Accounts.
Want to customize how the phone will notify you of incoming messages? Tap the Menu key while viewing any account's inbox, and select Settings. If you don't want to be notified of new messages at all, uncheck the Email notifications box. If you want to receive only a visual notification, an alert in your notification panel and a flashing of your phone's LED, set the ringtone to silent and make sure that the vibrate option is unchecked. You can set individual settings for each Gmail account on your phone: Just repeat these steps from any account's inbox, and the changes will apply only to that particular Gmail address.
Don't forget that your sent, deleted, and even spam messages are all synced by default, too. Press the Menu key and select the View labels option in Gmail to access all of your different mail folders.
Android's combined inbox
In addition to the Gmail-specific application, Android provides a general email app that can connect to multiple POP, IMAP, and Exchange accounts. Simply open the email app (found in the main system app launcher) and then follow the prompts to add any email account you wish.
You have the option of configuring multiple accounts within a single combined inbox. While inside the app, tap the Menu key and select Accounts. There, you'll see a list of all of the email accounts you've configured. You can tap on any account to open it individually, or you can tap on Combined Inbox to view messages from all of the accounts together. You can always add more accounts from this screen, too; just tap the Menu key again to find the Add account option.
The email app, it's worth noting, does not provide what's known as "push email." That means new messages won't show up instantaneously as they arrive, as they do with the Gmail app; instead, the app checks your accounts for new mail once every several minutes. You can define how frequently it checks each account by accessing the Account settings menu (press the Menu key while viewing any account), but the most frequent option is only once every 5 minutes.
Unless you're using an Exchange server, syncing your Android phone with Outlook can be a bit tricky. Google provides a full Outlook sync utility, but it's available only to paid Google Apps users.
If you're not using Google Apps, try GO Contact Sync. This free and open source utility syncs your Outlook contacts with your Gmail contacts. Make sure that you use the same Gmail account you set as your phone's default, and the contacts will automatically sync from Gmail to your phone. For your Outlook calendar, grab Google's free Calendar Sync utility. It syncs your Outlook calendar to your Gmail calendar, which also syncs automatically to your phone.
Using Google Voice
Your Android phone offers built-in apps for texting and voicemail, but you have a better option: Sign up for the free Google Voice service. Google Voice takes over your phone's text and voice messaging, and with Android everything is built right into the device.
Once you have a Google Voice account, open the Google Voice app, which likely came preinstalled on your phone. (If you can't find it on your device, try searching in the Android Market.) Follow the prompts to connect to your account, and you'll be up and running in seconds.