I never played SR1, but I've got the rulebook. (And if you want it too,
Stiggybaby is still selling it for a few bucks.) I did play SR2 for several years before SR3.
SR1-->SR2 was definitely the bigger jump. SR1 had a hosed damage system where any armor at all would practically let you walk through a storm of bullets. IIRC neither decking nor rigging were especially defined until
Virtual Realities and the
Rigger Black Book, and both of those systems were a lot more exotic than the current ones. (The VR rules in particular essentially forced GMs to write an entirely separate adventure in the Matrix, with rooms and everything. It was vivid but ridiculously time-consuming.) Those rules, alas, carried over into early SR2 until they were once again redefined in supplements. I also can't remember how magic worked, but I remember it was different. >8->
SR2-->SR3 was subtler. The biggest change was the skills system; SR1/2 used a "skill web" where every skill was related to every other one, and you could default up the chain
or drill into concentrations and specializations, and none of it related to attributes. It was actually quite elegant in its own way, but it was complex, and the SR3 skills chart is a lot simpler to understand. The only major combat change was the initiative fix (which a lot of street sam players cried havoc about). Magic in SR2 had a ton of dice pools which were all based on your skill ratings; SR3 simplified that, and also rewrote the metaphysics such that walking around with a focus was no longer a sign that said "HELLO! FIREBALL ME FROM ASTRAL!"
Other than that, there were some minor racial and chargen tweaks, but mostly SR3 was just an excuse to repackage the newer, mostly-better Matrix and rigging rules and clarify some things that were confusing to people. I've been telling my players for the SR3 campaign I'm about to start that if they already have SR2, they needn't buy SR3 unless they want to; I can just hand them a copy of the new skills table and they'll have most of what they need for chargen.
SR3-->SR4, of course, looks as if it will throw away virtually everything in terms of crunchy bits. It's the first time the core dice mechanic has changed. I make no judgments right now whether the change is better or worse; I've simply decided that I'm in no hurry to pick it up just yet, as SR3 is keeping me happy.