My wife is thrilled with the "new" iPad 2 she inherited when my new iPad arrived. I wanted to configure it for her so she could get up and running, so I thought I would just change a few settings here or there and give it to her using my settings. After about 30 minutes of tweaking, I decided to start over from scratch.
The reason is simple: there are just so many settings underlying the personalization of the iPad that un-doing them all was nearly impossible. It was much easier to add the things that I knew she would want rather than remove the things that she might not. Even the things that I knew I wanted to change, like Messages, left residual effects behind. For example, Spotlight had cached my iMessages, and continued to show them even after I had logged out of my iCloud account and into my wife's.
Long story short: start clean, sync back the apps and media you want to keep, and work from there. Here's how.
Backup your iPad and transfer purchases from iPad (optional)
Before you reset the iPad, you probably want to back it up and transfer all of your app and media purchases to iTunes. You can do this by connecting your iPad to iTunes and Control + clicking (or right-clicking, or two-finger-clicking on a trackpad) on the iPad in the devices list in iTunes. You'll get a contextual menu that lets you sync, back up or transfer purchases.
Of course you can always re-download apps and music from the App Store/iTunes Store, but if you have large applications it will be faster to sync them from iTunes.
You may also want to backup your iPad, just in case. Chances are you already backed it up before transferring yourself over to the new iPad, but it doesn't hurt to do it again.
All of your data from the iPad can be deleted very simply either from iTunes (by clicking the Restore button on the device summary screen, and then choosing "Set up as a new device" once it's wiped) or right on the iPad. On the device, go to Settings » General » Reset and choose Erase All Content and Settings. You will be asked to enter your passcode lock (if you have one set) and then confirm your choice.
As one of our commenters points out below, you can also issue a Wipe command using Find My iPad, if you're so inclined.
(Aside: you may have set the option to delete your data if you enter the wrong password 10 times. However, the iPad will start to introduce a delay after a certain number of mistaken entries, so that isn't a good way to reset your iPad on purpose.)
Once you reset, the iPad will reboot, and after a few moments you will see the initial configuration settings. One of these is the setup detail for an iCloud account. Even if you are planning to share App Store purchases, each person should have their own iCloud account/Apple ID. The reason is that Messages, FaceTime, and many other setting are specific to particular users. Also, more and more applications will start to be able to sync documents through iCloud, and you will most likely want those to be personalized.
The good news is that Apple provides you several places to enter different Apple IDs. For the initial setup, make sure to enter the Apple ID of the primary user of the iPad. If you need to create one, you can do it right on the iPad.
Sharing App Store Purchases
One Apple ID can be used on up to 10 "devices and computers" ("devices" here refers to iOS devices). It is very important to note that "[o]nce a device or computer is associated with your Apple ID, you cannot associate that device or computer with another Apple ID for 90 days." So you'll want to get this right the first time.
In most circumstances, the only thing you'll want to share an Apple ID for is App Store purchases. To change that Apple ID, go to Settings » Store and tap on the Apple ID. Then tap "Sign Out" to logout the current Apple ID from the App Store. Then log in with the Apple ID you have used for purchases in the past.
Sharing Calendars and Contacts
Sharing calendars is very easy. Just go to iCloud.com, log in with your Apple ID, and click on the calendar you want to share. Then enter the email address of the Apple ID you want to share your calendar with, and decide if you want to give them "View & Edit" access (so they can add, delete, and change events on your calendar), or just "View Only" access. (Unlike Google Calendar, iCloud calendars do not have an option for only sharing "Busy/Free" information without specific details.)
You don't need to do anything at all to enable that sharing on the iPad itself; that's done at iCloud.com.
My wife and I share our calendars that way, but we also want to share our contacts too. As you'd expect, we have not only family members in common, but also friends. If we are planning to meet at a restaurant, I'll enter the information into my iPhone, and it will sync to her iPhone (and iPad) too.
On my wife's iPad, I went to Settings » iCloud and turned off Contact syncing, since that connects to her account and is empty. Then I went to Settings » Mail, Contacts, Calendars, chose "Add Account..." and then select iCloud from the list of account types. I entered my iCloud information, and then only enabled Contact syncing.
Pro tip: When setting up contact sharing on my wife's iPhone 4S, I waited for the contacts to sync and then went to Settings » General » Siri » My Info and chose my wife's contact information.
Other Apple IDs
Here are some other places you'll need to enter your Apple ID:
Home Sharing: Settings » Video » Home Sharing
FaceTime: Settings » FaceTime
Messages: Settings » Messages
Game Center: For some reason, Game Center settings aren't in the Settings app, but if you launch the app, it will prompt you for your Apple ID.
(Note: iTunes Match under Settings » Music appears to default to the same Apple ID as in the App Store, which makes sense since it's linked to music purchasing)
Apple even suggests using one Apple ID for iCloud and one for the App Store as an alternate setup. Whether you choose to do this for your config is up to you, but if you want to avoid buying essential apps separately for your family's two devices (or more) then you probably want to use the same Apple ID for the App Store on all of them; you might want to turn off automatic download of app purchases to the hand-me-down device, though -- and the same with Photo Stream.
Another important decision is which Apple ID you want to use for Find My iPad. I associate all of our iOS devices and Macs with our main Apple ID (the same one used for making App Store and Mac App Store purchases). That way, we can have a single "console" for tracking down anything that goes missing, rather than having to remember a list of Apple IDs and their associations with specific devices.
Worth the effort
Resetting the iPad might seem like extra work, but I believe that it's well worth it. You may find that you and your spouse (or whoever receives your hand-me-down iPad) have very different preferences. Fortunately my wife and I both agree that Keyboard Clicks (Settings » General » Sounds » Keyboard Clicks) are awful, and the side switch (Settings » General » Use Side Switch To) on the iPad should be used for Lock Rotation, not Mute.
If you passed your previous iPad along to a family member, let me know what other settings you customized for them. We discussed sharing an iPad between spouses back in 2010.