I think I should replay SC1 and Brood War, or at least I should go on their Wiki pages and read about the story for those two. I played and completed both (SC1 and BW) long... oh so very long ago. I remember about 5% of the story, however (but as I said I think I'll just at least Google that info at some point).
You should definitely play SC1. The game aged well due to its excellent 2D sprites and gameplay is superior- there's a reason why SC1 was THE global eSports game in 90s & 2000s.
Not just for story either. The gameplay is fun. It's not nostalgia talking because I played SC1 mere few years ago and I still had a blast.
As for the story of SC1 & why it was so good.
1. Jimmy Raynor was retconned to this cool rogue guy with pretty hair & cool tats. In SC1, he was a backwater marshall that was bald. He was your average guy who rose to the challenge. And when Kerrigan wronged him & murdered everyone, he VOWED to kill her. "It won't be today, or tomorrow, or even with an army behind back. But it will be me that's going to kill you."
2. Kerrigan's charact development was awesome too- her Ghost program under Mengsk & how she met Raynor under Mengsk. There were some flirty banters, but it never devolved to some cheesy romance. Then she was abandoned by Mengsk and captured by Zerg. Zerg infested her as who she is today. You get to play fun Zerg missions where you guard her in her chrysalis form. Then she hatches & wrecks havoc. Just few missions ago, you played as Terran, controlling Raynor. But suddenly you're playing Kerrigan as Zerg against Raynor. Very very well done & lots of fun.
3. You get to see the origins of Dark Templars and how they were banished by High Templars. And how badass Zeratul & Tasaddar are that united them together against Zerg (Overmind).
4. The lore of SC1 is a true galaxy politics, war, betrayal, & unity, all backed up by solid character developments.
5. Just go play SC1 & BW, you won't regret it. It'll be an awesome 2-3 weekends for you. The in-game voice acting is absolutely superb.
In SC2, suddenly Raynor is a pretty boy with full head of hair. They have really awful characters you forget about them- Horner, and lol that meathead Tychus. SC2 Wings of Liberty's ending was awful with that corny white knighting of Raynor, saving Kerrigan.
Zerg is a race that composed of assimilated creatures of various planets (zergling, hydra, mutalisk, etc). But suddenly they're all from 1 planet in SC2. They totally gutted everything.
SC2 is awful story-wise. If you're enjoying the gameplay, then that's fine. It wasn't bad for me either.
Go play SC1, you'll love it.
Some quotes from SC1 (at least early, top of the page)
http://www.imdb.com/character/ch0041709/quotes