How can I learn to make mesoscale/synoptic analyses of major weather events?
You're definitely on the right path with wanting to dig into raw data yourself.  RAP has always been a pretty solid source of useful basic data.

But in terms of making sense of the data, it indeed will take some practice. You can feel like you're not getting it at all, but suddenly pieces will start to click, much like riding a bike.

Basically, you need to start building from those that know these things.  As such, one good insight can be in reading NWS forecast discussions, which you can find among the rich data at kamala.cod.edu or in one big set at this NWS link.

But, at least in terms of storms, I think the place to start has to be the same place that the CWG article relied upon: the Storm Prediction Center.  Specifically:


Convective outlooks will explain the day's weather picture and large key features
Mesoscale discussions offer insight into the finer details that make all the difference, ones which even the most seasoned meteorologists may not always recognize.
Then the mesoanalysis tool allows you to dig more into the data yourself and see overlays of a deep variety of parameters.  But the tool that may be right down your alley may be the composite map generator tool, which allows you to combine just the noteworthy parts of the key elements onto one image and see how they line up.
And then the Severe Weather Events Archive allows you to dig into a great deal of past events, and the Outbreaks Browser lets you see the bigger events in a very sharp, graphical way.


And there's plenty of other great things there too; climatologies, publications, education pages, etc.  Now, obviously the majority of this information focuses most directly on severe weather setups (though the SPC does actually also issue winter weather mesoscale discussions [the WPC now does heavy rain MDs, and of course the NHC does the main discussions on tropical systems]), and only regard US events.  But these tools can really help even those interested more in other areas and topics get a better understanding of the bigger picture, which sync up well into just about every topic of synoptic and mesoscale meteorology, regardless of location.

Now, in terms of helping you personally narrow down onto the key elements, so that you could make your own composite map... the classic ingredients necessary for thunderstorms (also see here) can be a good starting point:


Instability... the predisposition of rising air to continue rising.  Basically warm (and moist) surface air and anomalously cold air aloft.
Moisture... high dewpoints create clouds more readily, form them at lower heights (important for tornadoes), and produce more precipitation/energy for storms.  Plus it helps the instability.
Trigger... something to get the air moving to rise so that the ingredients can be realized.  There are a WIDE range of possibilities here.  Important ones for storms include fronts, the dryline, vorticity advection aloft (incoming vort maxes you see spinning around on water vapor imagery), favorable jet streak quadrants, strong temperature advection (e.g. low-level jets), outflow boundaries, seabreezes/lakebreezes/riverbreezes, gravity waves, diffluence aloft, terrain/orographic lift, horizontal convection rolls/cloud streets, isentropic upglide, and differential heating boundaries.  Suffice to say there are a lot of options in play, though some are more common than others.  But you'd be surprised how quickly you can get used to finding them with practice.  (And it's really a pity there isn't greater information/graphics/videos/explanations on some of those topics... perhaps some of these will eventually work their way here to be questions that can encourage a better, clearer reference to form)


And finally, to really ramp up severe weather, and also primary to tornado threat, you add in ingredient four...


Wind shear... different winds at different height.  A combination of how much directional difference there is and how much speed difference.  Significantly impacted by upper level jet streaks, low-level jets, and the direction of the surface flow is.


This may sound like a ton.  But taking a second look at that Accuweather graphic, it's those same ingredients:


So with those basics to start at, the pictures have a pattern to them.  And then when you add in the finer details discussed, you can really find the deeper things you won't on the more basic composites.  You can see what really DID make all the difference.  In the end, it's an investigative science, a continual game of making and refining theories, and it's something you only get better at with practice.  Even top meteorologists still don't always have the answers; I don't know whether anyone has come up with a perfect explanation yet for why the supercells on Sunday in Central Florida didn't produce prolific tornadoes.  Forecasts do still fail.  And so we're always trying to better clear up the details of each past event, so that we'll be better at the next one.

A couple final links... it seems like this journal article might be fairly interesting and approachable for you... and then websites like Stormtrack can be a really great asset, as that's where quite a few chasers/meteorologists often congregate to discuss big weather events.  The more time you spend looking at this stuff, engaging in conversation, challenging the reasoning you/others have, and learning, the better you'll be able to form a clearer picture of what's going on.  A much better picture than any news article will ever give!