If you only want to change the account and don't need to merge the data from the two accounts, just go to Settings>iCloud on your phone, sign out, then sign back in with the other account ID. (If you want to keep your photo stream photos, save them to your camera roll before signing out.) If it's the account on the phone that you want to use, go to System Preferences>iCloud on your Mac, sign out, then sign back in with the ID your phone is using.
If you want to merge the data, and want to use the account that your Mac is using, start by saving any my photo stream photos that you want to keep to your camera roll on your phone (unless already there) by opening your my photo stream album, tapping Select, tapping the photos, tap the share icon (box with upward facing arrow), then tapping Save Image (or Save to Camera Roll if you aren’t running iOS 8). If you are syncing notes with iCloud that you want to keep, you'll need to open each of your notes and email them to yourself so you can later copy and paste the text into new notes created in your new account. Then go to Settings>iCloud, tap Sign Out (or Delete Account if you aren’t running iOS 8), choose Keep on My iPhone and provide the password to turn off Find My iPhone when prompted. Then sign back in with a different ID that your Mac is using and choose Merge to merge the data from the old account with the data in the new one.
If you want to merge the data and want to use the account that your phone is using, go to System Preferences>iCloud, sign out, then sign back in with the ID that your phone is using. Then go to System Preferences>Internet Accounts>iCloud and add the account that your Mac was using back as a secondary account. Then you can drag and drop contacts, contacts and notes from the old account to the new one. If you need to move a calendar from the old account to the new one, export it from the old account as an .ics file (File>Export>Export), then create a new empty calendar in the new account by the same name (File>New Calendar>iCloud), then import the .ics file you exported to the new calendar (File>Import). When you're done, delete the secondary account.