You're not going to get much understanding of the plot between 2&3 as any connection requires SR2's Downloadable Content when helping Eric Gryphon might be required. Though I guess there's always the radio commercials in SR3 that can easily state this.
The only other connections between 2&3 that I remember had to do with Voodoo as mentioned during the Zombie Mission, a Septic Truck that was mentioned during Professor Genki, & the death of Jessica during a STAG speech.
As for the connections with 1&2 there is only the one dealing with how SR1 ended, a mechanic, & whoever returned from the previous. Beyond that any thing else I had found is that 3rd Street Speeches play homage to SR1 & how someone dies in a way you would have. (Though I guess that would be 2 different ways.)
There's also always the spoilers from Kia about your former homies who have died.
But ultimately I'm still left with so many questions about Dex, Dane Vogel, & Eric Gryphon that from what I can tell from SR4 may (or will) go unanswered.
Finally Saints Row 1 is only available on Xbox 360 if you can buy a preowned copy. Saints Row 1 also lacks mission checkpoints so if you fail you start back at the beginning. And be prepared to earn respect before advancing the story in either Saint Row 1 or 2 if you buy them.