Was planning on playing the series, but honestly wanted to try out IV, because everyone talks about that one being really good. Will I skip a bunch of story, or does it update you on what's been going on? I'd hate to miss out on stuff.

You could skip 1 and 2, but should probably play WC3... this is were the full acted cinematics started, and you'll meet many of the characters present in WC4, played by the same actors; it would be a shame to miss that, especially if you already have it (and if not, is free on Origin right now).

There is also some background on some characters from the two older titles, but the story there was quite light, and you can easily put yourself on par by reading a wiki

Also, the flight engine in chapter 3 is, again,where they started using the new, real 3D, one; it's not that different from chapter 4 and it is still enjoyable.

So, I would personally play this as a trilogy: WC3-->WC4-->Prophecy.

I'd recommend playing from 3 onwards. I preferred the story in it, but then again I did play the hell out of Wing Commander 2 way back when, so I was a lot more invested in the war against the Kilrathi.

3 onwards and you should be fine, this is how I played the series back in the day... though the whole series should be discounted if money is a factor. Going back and playing 1-2 is on my list to do to get the full breadth of the series (I first saw them on a friend's computer, but never played through them myself).

(I know, it is late, but better late than never - and besides, I think that you are not the only one with this question)

Well, you should. Otherwise you will not be able to understand a lot of background information, including the character relations.

Relations between characters were built over four first games quite carefully - without previous games you will miss A LOT. For example, without playing Wing II you will never understand, why is the relationship between Tolwyn and Blair so tense and I could go on for several pages.

At the very least, find some good playthrough videos on youtube and watch them (as I realise, that Wing I is probably unplayable for todays players - heck, i can not play it, it is SO OBSOLETE even for me, who played it when it was out back in the early 90', Wing II is slightly different beast as it is more story driven and has more logical controls, but you must be able to stomach its graphics and sound - which, I realise again, may not be possible) .
Wing III onward should be OK, as its references to previous games were "butchered" in order to lure newcomers. But Wing IV returned to the standard set by older games and sort of requires the knowlegde of the entire continuity again.

Oh, and don't bother with Prophecy - that is not a Wing Commander game... ;-)

And I almost forgot - Armada is partially part of the main continuity as well... (while it is not essential, it is good for understanding some background information from Wing III... well, here we go again, I guess...).
Privateer and Academy can be safely ignored (+/-).
And Privateer 2 does not even belong to this universe.

I've played all of the Wing Commanders excluding 4. You do not need to play 1 and 2 to understand 3. The story is very fun and engaging, but not very consequential.

All you need to know is this:
In Wing1 you were the hero on board of the Tiger's Claw, didn't really bond with anyone all that much, got lots of medals and a promotion to Colonel.
In the intro of Wing2, the Tiger's Claw is destroyed and you're blamed for it. That's why Admiral Tolwyn gives you such a hard time in Wing2 and Wing3. You spend the game cleaning your name and wind up the hero again. Along the way you start a relationship with Angel, who is captured in the Intro of Wing3. In Wing1 and Wing2 you're also friends with Paladin, who make brief appearances in Wing3. He was a pilot in Wing1 and a secret agent in Wing2. The last person who is known from earlier games is Maniac. He is a hotshot pilot that you don't like.

The end. Have fun playing Wing3.

WC4 is definitely the best of the saga IMHO, but the first ones are still worth playing. And it's true that, while the story of WC4 doesn't depend much of the story of WC1-3, the characters and their relationships do - the relationship between Blair and Admiral Tolwyn or between Blair and James "Paladin" Taggart is built slowly over the first 3 episodes, and plays a role in WC4 events too.