The games are decently designed to allow you to start the series at any point.
I'm trying to figure out how to say this and be as spoiler-free as possible, but:
The first game has you hunting down a rogue agent. This mission exposes you to the big bad of the series, but has a strong focus on your mission.
The second game has you investigating why human colonies keep disappearing. In that process, you learn more about the big bad, and their threat is tangibly increased. But you still have the more pressing mission of disappearing colonies.
The third game is when the big bad is finally an imminent threat, and Shepard has to deal with that.
Mass Effect 2 and 3 have combat systems that are very similar, but a fair bit different from ME1 (at least in experience of them). Mass Effect 2's main cast is largely independent from ME1, and the only returning crew members are ones that were actually optional in ME1, so it's not an issue that you don't really know them. If you don't transfer an earlier save, THEY won't be close to your Shepard (more like: "Oh hey, yeah, we met when you were tracking down that guy" than "BEST FRIEND!")
So it's not really a jarring transition.
As for 2 to 3, you can absolutely start with 3. But it's kind of like starting with Return of the King. I mean, yeah, you'll get what's going on. And they'll give you enough info you'll get it well enough. But you won't really understand. They actually added one companion in ME3 specifically to help new players get into the game, because he's new to the squad. So the party banter is primarily about giving him a history lesson in the form of them telling him war stories. And he'll ask for information in conversations that he wouldn't have known.
So yeah. Like any series, the latter entries are better if you played the earlier ones. But they do a very good job of making them accessible, in my opinion. The worst part is that you don't get the effect of the cameos and returning characters.
I should also note that the game defaults to primarily Renegade ("bad") for discussing any past events. So if you want to play Paragon ("good"), you'll still have to deal with the ramifications of some significant past Renegade choices.
Anyways, you all are horrible, @#%^ed up people