Mass Effect 1:
The main plot didn't make much sense. Saren attacks a human colony and risks his Spectre status to obtain a Prothean beacon he already possessed (Virmire) to find the location of the Conduit that he didn't need, to gain access to the Presidium he and his Asari commandos already had full access to.
As much as people like to rip on the plots for the sequels, ME1 had the most nonsensical of the bunch.
Because ME1 was such a fun game I was able to overlook those plot holes and just enjoy the ride. And it is still one of my favorite games of all time. But if I were to be honest and list things I'd change in the series, ME1's main plot would be a good place to start. In retrospect perhaps Saren shouldn't have been a Spectre (I wonder if that was the case in an early draft of the script) or there should have been some compelling reason for him needing the Conduit despite being a Sprectre. Virmire also probably shouldn't have had a Prothean beacon and the communication with Sovereign should have occured through other means.
Mass Effect 2:
The Council reverting back to air quoting Reapers after being attacked by one in the finale of ME1 and acknowledging their existence in the final minutes of the game. No doubt Bioware did this to force Shepard to work with Cerberus, but it came at the cost of making the Council seem like ostriches with their heads in the sand. And it wasn't necessary.
Instead they could have had the Council continue to accept that the Reaper threat was real, but disagreeing with Shepard about where the war effort should be focused. Have the war with the heretic Geth still be raging at full steam, with the Council both disagreeing that the Collectors are linked to the Reapers and that they should be the main focus, rather than the Geth. Shepard's disagreement and split from the Council then is not because they aren't taking the Reapers seriously, its because they don't see eye to eye on where resources in that war should be allocated.
Also the number of organics harvested to complete Reapers should probably be lowered to thousands instead of millions. Having hundreds of thousands of people vanish *should* raise eyebrows and lead anyone to suspect Reaper involvement. Instead have isolated colonies of a few hundred or thousand here and a thousand there disappear, so that it is still reasonable for people to blame pirates and Batarian slavers, or to suspect that the Collectors are nothing more than aggressive alien race of slavers. Lowering the requirement of people needed to build a Reaper also doesn't necessarily increase their construction rate, as they are still highly advanced starships that would require lots of time and resources to build. Have most people harvested by Reapers processed into husks or any of the assorted Reaper goons instead of Reapers themselves.
Drop all references to humans being special, because we aren't. We are one of the least genetically diverse species on our own planet, so having the Reapers being interested in humans because of genetic diversity is a bit silly.
Some casualties should have been unavoidable on the Suicide Mission. A perfect run should have been one where Shepard's squad sustained the minimum possible number of casualties, but the mission itself should not have been without loss.
All characters should have had at least one outfit that was proper combat armor with an optional helmet, and looked as if it would protect against the vacuum of space and other hostile enviroments. That also should carry over into the next game.
Mass Effect 3:
The Reapers should not have arrived in the Milky Way by hitting the gas pedal and flying there at conventional FTL. That rendered the plot of Mass Effect 1 more nonsensical than it already was, and it also raised questions about the plot of Mass Effect 2. Why did the Reapers need the CItadel relay again? If they were capable of arriving at the Citadel by coventional FTL, wouldn't it have been smarter to do just that instead of bothering with the Sovereign/Saren scheme? They'd have taken the galaxy by surprise just as they did with the Protheans, had Saren/Sovereign not exposed themselves.
Instead the Reapers should have had a backup plan to their backup plans that didn't cause problems with the plots of previous games. The Batarians and the Leviathan of Dis provided an easy out for the writers. At some point during the course of the events of Mass Effect 2 the Batarians begin secret construction of a Mass Effect Relay meant to link back to their home system. It is completed at the start of Mass Effect 3, with the Batarians announcing its existence to the galaxy with their usual arrogance, claiming it is a demonstration of Batarian technological supremacy. Unfortunately for the Batarians however both the original idea and the plans for the relay were introduced by the Reapers via the indoctrinated scientists, engineers, and military officials who had come in contact with the Leviathan of Dis. When the Batarians activate the relay, instead of linking with their home system it realliigns itself to a relay hidden deep in Dark Space, the Reaper fleet jumps through, and the extinction cycle begins.
Alter the Rannoch arc so that the Quarians aren't foolishly going to war with the True Geth while they know the Reapers are coming. Instead have them be warring with the Heretic Geth only, who of course would have to survive Mass Effect 2 which means Legion's loyalty mission would need to be altered. The Quarians then are engaged at war with a Reaper ally and their leaders don't look completely incompetent. Have the Heretic Geth manage to wrest Rannoch away from the True Geth at some point prior to ME3 and succeed in rewriting a large portion of the True Geth, turning them into Heretics, and putting the Quarians in danger of being overwhelmed. The Rannoch arc then is focused on either 1) annihilating the Geth entirely or 2) aiding Legion in undoing the Reaper/heretic rewrite and adding the restored True Geth as allies. Rannoch would be about ending one front of the Reaper War to free up Quarian (and possibly Geth) forces for use on other fronts, not about ending a stupid and unnecessary war that originally had nothing to do with the Reapers.
Cerberus should play a less prominent role and the reveal that the Illusive Man is indoctrinated should have been a twist, not something you saw coming a mile away from the first time you encounter Cerberus troops on Mars. Instead Cerberus should have returned as an ally of sorts initially, with the betrayal coming as a surprise at Thessia. Shift the coup to post-Thessia. Give Jacob and Miranda character arcs related to rallying unindoctrinated segments of Cerberus against the Illusive Man after the reveal. Have the role of indoctrinated goons at Mars and side missions be filled by indoctrinated troops from the Terminus and Batarian space.
The ending: Discussion of that would really need its own thread to discuss in detail and has already been done umpteen times. But I think where Bioware mostly went astray in the original endings was is in having the cost of victory be the loss of the galactic civilization that Shepard was trying to save. There is nothing wrong with a bittersweet tone to an ending, I just think Bioware would have been better served by having the sense of loss come from a casualty or two in the end run than by the destruction of the setting itself. On that note I wish the end run had involved another ME2 style Suicide Mission with all surviving ME2 squaddies joining for the final mission. I'd also scrap the Catalyst, have the final encounter be with Harbinger, and have the function of the Crucible by less mysterious. Thats not to say that the effects of the Crucible can't be space magicky, just that if you're going to use space magic, it has to make sense within the setting. (like biotics, element zero, Mass Effect physics, ect.)
Anyway, here was my attempt awhile back at an ending for Mass Effect 3. (TL:DR warning)