How important is it to backup a new or refurbished iMac right away? Can it wait? Do I really need to backup my iMac at all? How likely is it that my iMac will crash or have hard disk issues? Any other advice you can give on backup would be great.
If you have no data you need then you don't need to back up. Once you have pictures, music, documents, etc. that are vital you should be backing up regularly. With Time machine there's really no excuse.
From a statistical point of view you are less likely to experience a hard drive failure the newer the hard drive is, however this doesn't mean it cannot happen. Some drives last for 10 years, others 10 days. Bottom line is, if you value your data, you should back it up. 😎
Solutions? I recommend evaluating Time Machine, Carbon Copy Cloner and SuperDuper. As for external drives, I have always been a fan of Seagate drives and their 5-year warranties, and Lacie makes a solid product as well (keeping in mind they do not actually manufacture drives themselves).