Mass Effect Series – Is There a Canonical Commander Shepard?

As I mentioned in my comment, the ME Wikia has a really detailed storyline article that captures all the games, DLC, and books, but it's a long read, so here's the short rundown:

The Mass Effect series is the story of Commander Shepard, who is essentially a special forces soldier who gets tapped to be the first human to join an inter-species galactic investigative/special projects team called the Spectres. he's given a prototype ship, called the Normandy. Humans are relatively new compared to many other established races in the galaxy, and are sometimes viewed with disdain.

In the first game, he is tasked with investigating a Prothean (ancient spacefaring race that died out long before the evolution of humans) artifact. What he discovers is another of the Spectres (Saren) has gone rogue and is working with the Reapers (another ancient machine-based spacefaring race) to bring the Reapers back online and wipe out all sentient life. He eventually stops Saren and beats back a Reaper scouting attack. He learns along the way that periodically the Reapers return and harvest all sentient life in the galaxy. The Protheans managed to stall the next attack before they were almost completely eradicated, in the hopes that during the next cycle sentient life would be prepared enough to stop them. However, the galactic council refuses to believe that the threat is as serious as Shepard makes it out to be.

In the second game, Shepard is killed early on, but is brought back to life by a group called Cerberus, led by the Illusive Man. Cerberus believes that humanity should be taking care of itself, and not listening to the galactic council or trusting other alien races. It's heavily implied that Shepard is special or unique in the battle against the Reapers. Shepard reluctantly works with Cerberus in order to further investigate the Reaper threat, and fights the Collectors, who are Protheans that have been "assimilated" by the Reapers. In the end, Shepard takes the Normandy through to a Reaper stronghold and discovers that the Reapers are working to assimilate humans as well. He manages to destroy the prototype human/Reaper hybrid, and escapes knowing that a Reaper invasion is imminent. His decisions may strain his relationship with Cerberus, however.

Still, no one believes the magnitude of what's coming, and the best he can do is continue to prepare for the Apocalypse that he knows is just around the corner...

You miss out on a few things, I'd only consider the interactions with Wrex as major:

1. BDTS DLC for the first one allows you to have some war assets if you took the paragon path.
2. Conrad Verner needs to have survived in both the first and second to show up as a war asset in the third.
3. You have to have saved the Rachni queen in the first to gain her as an asset in the third. (The fake rachni queen that's subsituted will betray you).
4. You will miss out on Wrex in general as he's now dead.

There are more but most of them are simply war assets etc.