My contacts did not transfer to my new iPad. Everything else did.

Did you include contacts in the backup or were you syncing contacts with iCloud? If you were syncing contacts with iCloud, you never to turn that on again in the settings so that your contacts will download from the cloud. That is achieved by going to settings>Your Name>iCloud>Apps using iCloud>Contacts>On.

If you are syncing contacts through your email account, you have to make sure that is turned on in the mail settings for your account.

iCloud is enough to share contacts to your iPad. You should follow what Demo says here to turn on Contacts both on iPhone (or your old iPad) and your new iPad. To check whether youer contacts have been uploaded to iCloud, you can go to https://www.icloud.com/ and select Contacts to view your contacts.

I prefer to saving my contacts on computer so I used another application. You can refer to this guide how to sync contacts from iPhone to iPad.