Yes, the update is necessary for security and on-going functionality.
The backup recommendation has to do with the fact that the 3.6 update is going to make a modification to your database. So, making a backup beforehand leaves you with some fallback options just in case something might go wrong there. Overall, the backup you need there is your wp-content folder where your database will be modified.
I used to be skittish about updates, but no longer other than to do as you have done by reading the forums a bit to see what issues might be popping up for others…and like I believe wpYogi has said, this update is proving to be a very smooth one.
Disable ^all* plugins, and be sure to do that properly if you have a cache or security plugin that needs a bit of prior attention there. After that, switch to the Twenty Twelve theme and then do the update. It will not take long, and you will be shown what is going on. After the update has completed, re-activate your plugins and then your theme and give things a few sedonds to settle in and all should be well…but do not panic if something does not seem right. Rather, just come back here and one or more of these fine folks will walk you through whatever might need to be done.