Always presume the whole machine is going to be trashed beyond repair by this process. Prepare for the worst, hope for the best. If you don't have a test machine, you can build one with a tool like Vagrant. Once you have a procedure that works, repeat it on your production system.
Ruby 2.1.1 is the current version. You'll also want to look at upgrading Rails itself to avoid a bunch of nasty vulnerabilities. 3.2.17 is the version to target here.
RVM does make it easier to upgrade things, but you'll also need to upgrade your launcher (e.g. Passenger) to use the newer Ruby version. Yes, it is possible to back out, but this is not always convenient.
One trick that might help rescue from disaster is checking your /etc
directory into a local Git repository. This allows you to rollback any configuration changes you make, as well as see what changes you've actually made through the course of your upgrades.
Any change to the base Ruby version does require re-installing all gems. If you're using Bundler or an automated deployment tool this should be fairly automatic.