If you create and use a wholly new (not amended) Apple ID, you won’t be able to transfer anything you have purchased from the iTunes & App stores to the new ID, but you can transfer your iCloud data (calendars etc) with a little patience.
iTunes & App Store.
You may want to think about continuing to use the previous Apple ID for the iTunes & App stores (as well as iBooks), so that you can continue to use your purchased content without a problem and just use the new Apple ID for iCloud, Messages and FaceTime.
If you do use your new Apple ID for the iTunes and App stores, you will be able to authorise and play previously purchased content from the iTunes store on a computer, but may encounter difficulties using this content on an iOS device.
Whilst you won't lose apps you purchased using the previous Apple ID immediately, you will no longer be able to update them.
If the previous Apple ID is going to continue to be used by a family member who is part of a family sharing set up which will include your new Apple ID, you will be able to share the previous purchases as well as any further purchases they make under that Apple ID. Be aware though that when using Family sharing, unless each family member has their own credit sum from iTunes gift vouchers, the organiser of the family share set up will pay for any purchases made by any family member.
iCloud
Moving your iCloud data from one Apple ID to another is achieved more easily on a computer than it is on an iOS device, indeed certain aspects of the move can only be done on a computer. However, whichever device you choose to move your data on, you should do so by turning individual iCloud services off one at a time and not sign out of the account until all services have been turned off and dealt with. (Settings > iCloud on a mobile device and system preferences > iCloud on a Mac)
If you have purchased extra space for iCloud under your previous Apple ID, this won’t be transferred when you change the Apple ID you use and you will need to re-purchase the required space under the new Apple ID.
iCloud Drive. Save data from your iCloud Drive to a computer before turning off this service. You can use the Finder on a Mac or through iCloud.com if you use a PC. Simply move the data to your new Apple ID once it is set up.
Photos. Make sure you download and keep originals when prompted when you turn iCloud photo library off, you want the photos to remain on your device and merge with your new library when you set up your new Apple ID.
If you use photo stream be aware that there may be more photos on your device than there are in the cloud, photo stream in the cloud only keeps up to 1,000 photos for 30 days, if you have older photos or more than 1,000 on your device you may want to copy them to a computer before turning photo stream off.
If you have created any shared albums you will need to create them again once you have set up your new ID, any photos that anyone else as added to your shared albums should be imported to your camera roll/library before you turn off shared albums.
Mail. Once you change your Apple ID, you will have a new iCloud email address. So that you don’t miss any mail which is sent to your old address, you can arrange to have it forwarded to your new address (or any other address), you can set this up at iCloud.com > mail > preferences > general.
Contacts. You will be given the opportunity to save contacts to your device when you disable this service, merge them when prompted when you re-enable the service after signing into your new Apple ID.
Calendars. You will be given the opportunity to save calendars to your device when you disable this service, merge them when prompted when you re-enable the service after signing into your new Apple ID.
Reminders. You will be given the opportunity to save reminders to your device when you disable this service, merge them when prompted when you re-enable the service after signing into your new Apple ID.
Safari. You will be given the opportunity to save Safari Data to your device when you disable this service, merge them when prompted when you re-enable the service after signing into your new Apple ID.
Notes. You cannot save notes easily. On a Mac you can drag them into a different account and then drag them back once you have set up your new ID, but on an iOS device you will essentially need to copy and paste them into new notes when you have set up the new Apple ID, but you will need to share them before you turn of notes for the old Apple ID, either by messages, mail or similar.
Back up. You can’t save your existing back up you will have to start a new one once you have set up your new Apple ID.
Keychain. You will be given the opportunity to save passwords and credit cards to your device when you disable this service, merge them when prompted when you re-enable the service after signing into your new Apple ID.
Personally, I'd have kept the old ID and set up a mail rule to have mail from that person automatically deleted, that way you'd never even know you were getting any or ever see it.
