The recommended way is to use the Update Manager (mintupdate), because you'll have a "update pack information" button who lets you see what update pack you're currently running, if there is a newer one available, and special information (like adjustments, minor bugs, how to deal with some questions..)
But technically you can use "apt", "aptitude", "apt-get", "synaptic" or "mintupdate"
At least check the UM and said "update pack info" button before doing anything else. It will tell you what update pack you are configured for.
LMDE uses their out repos to filter the stable packages from the Debian testing repos. They have found that the easiest way to do this is build the repo slowly.
That is fine but what about security or some other urgent package upgrade that may be needed? To get around this problem they change the the repo name when building the new update pack, leaving the old repo (previous update pack) intact and working.
You can get the new repo addresses by checking the web site and manually edit the sources.list. It is easier to leave the auto settings for checking for updates alone in UM and it will edit the sources.list for you.
I would not let UM do auto upgrades. At least make it ask before doing so. That way you can check these forums incase there is a problem (doubtful but possible).
Apt-get is a good way to actually do get you package upgrades. It is actually the backend that UM uses for most of what it does. The difference is you get a lot more information as apt-get is running and can see if there is an obvious problem.
If you do have a problem with an update pack upgrade you will have more information to deal with the problem,
I use Debian testing or Sid as my production OS. I have LMDE (Xfce) on here to keep and eye on it as I think it is a great distro for noobs.
That being the case I only use UM to do update/upgrade cycles as that is what a noob is going to probably do. I have had no trouble at all with the system. Works pretty slick.
Dell XPS 420 Core2 Quad Q 6600, audigy5.1, Radeon HD 6450 - currently 4 320Gb HDD, Debian Squeeze for secure use, Debian testing for daily use, Debian Sid for fun.