Do you have access to the old Mac? If so, move your iTunes folder and all of its content to the new Mac. Then when opening iTunes, hold down the Option key and select your old library.
If you don't, your iOS devices will need to copy their contents to iTunes. You can achieve this by plugging them in and right-clicking them and hitting 'transfer purchases'.
What are you trying to sync? If it's not media like music or videos, you don't really need to sync them anymore. If it's your contacts, just turn on iCloud on the Mac and iOS device and your contacts will be synced without iTunes.