I'm 3 weeks into my 1st Linux (Mint) install and am wondering what methods and options are available for backing up the system and then recovering it. I've used Windows for years and am aware of methods for that operating system but have no idea what is common, popular and/or "best" for Linux.

You don't mention what desktop you're using. If it's KDE then I'd certainly recommend kbackup. If you don't care about a gui frontend then you could look at rsync. There's also a gui-ed version of rsync called grsync.
jdk

You'll (hopefully) quickly come to the conclusion that there is rarely a "best" option in Linux - but always plenty of contending opinions to fire up a debate ...

Best to always try to separate out what is current, and what is no longer developed (deprecated) - try to stay away from the latter for least grief.

Are you looking for a way to back up and restore files, or to restore the entire system at once?

There are solutions for both, but not the same solutions: ie the best answers for a full system backup are not best for restoring one file or folder.

There are 'clone' or 'image' backups that can record and replace your entire machine or only major partitions. There are backup solutions that allow for point-in-time recovery of data. In fact, there are so MANY excellent tools that it is difficult to know where to start!

First, can you refine your question to indicate what level (or levels) of backup and recovery you require? Are we talking data security, full system disaster recovery, something between, or multiple levels?

One thing I would suggest, if you have the spare hardware to support it, is to TEST whatever backup solution you choose. This is as much to test if you yourself know what to do as to test the software you are using. Pick a totally different computer, with a clean hard disk, or one that you don't mind overwriting, and see if you can bring up your backup on this test system. You will be using "Live CD's" and such, since you are working from a blank computer during testing. A "bare metal restore". It's a valuable exercise to perform, so you know what to expect, and can learn how to handle any problems under a no-pressure scenario.

I have not had any problems restoring things from Linux-based backups before. However, a long time ago when I used to use Windows, it was quite common for people to happily back things up right and left, and then when the time came to restore something, it didn't work! The problem is, many people have backup experience, but when you ask them if they have restore experience, the answer is, "Well, uhm, no, I've never tried that part." You SHOULD try that.

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Hi Negative_Space
I think Clonezilla is the best Backup and Restore program for Linux. I have used it many times and never had an issue with it. I burned it onto a CD and do complete backups as I do installs of various programs etc. I use the 64 Bit version and it seems to work fairly quickly. I use an External Disk to keep the various backups on, to make it easier to keep track of. I have restored many times without a hitch.