Do not use your new ID. Items cannot be transferred and it can be a real nuisance trying to manage multiple AppleIDs. Change your e-mail address for the old ID as instructed by Rysz. I have not put myself into this situation but you may find that now you have set up a new ID using the address you wish to use it is blocked for changing the old Apple ID to this address and you may have to dome some fiddly three way changing with a junk email address. I don't know.
Ideally transfer your old library to your new computer rather than re-downloading:
Moving a library: Quick answer if you use iTunes' default preferences settings: Copy the entire iTunes folder (and in doing so all its sub-folders and files) intact to the other drive, hold down the option/alt key (shift on Windows) while starting iTunes, and when requested to select a library select the iTunes Library.itl file in the copied folder. https://support.apple.com/HT201596
When this is to a new drive/computer put the copied iTunes folder in the default location of Macintosh HD > Users > *User Name* > Music (or Home > Music) then you don't even need to start with the option key held down, iTunes will automatically look for it there. (Make sure there isn't anything already in the iTunes folder there that you want to keep since you will be replacing it with the one you are moving.) Remember to authorize the computer in iTunes to your Apple ID if you use this to sync to Apple mobile devices or play DRM protected media (e.g., movies).
Certain items used in iTunes related activities are not stored in the iTunes library folder but in special folders. If you are moving a library to a different computer you may wish to copy these too:
- Apple Music cache files: http://dougscripts.com/itunes/2015/09/new-flush-apple-music-cache-files/ - ~/Library/Caches/com.apple.iTunes/SubscriptionPlayCache/ folder, which is sort of like an “iTunes Media” folder for Apple Music
- iOS device backups, various preferences files: https://discussions.apple.com/thread/3285925