Coincidentally, I’m in the middle of a legendary edition playthrough myself and had a similar reaction to replaying ME1. I’ve played through it before, so I was not concerned about spoilers and the like, but nearly every guide I found online about relevant side missions assumes you’ve played it before and is thus riddled with story spoilers. So, I’m going to try and adapt this spoiler-filled list into a non-spoiler version of side content stuff that carries over (lots of stuff from the main story carryover too of course, but those are not really missable):
A decision related to Tali’s side quest
Conrad Verner / The Fan
UNC: Rogue VI
Bring Down the Sky
‘Minor’ Consequences (usually just a stray mention in a dialogue our email in later games, all probably skippable but, as mentioned, you do need some paragon or renegade points for an event later. But note that I’m pretty sure that some of these missions have names that change during the course of doing them so it might be hard to find them without referring to a wiki.):
Citadel: Asari Consort
Citadel: Doctor Michel
Citadel: Reporter’s Request
Citadel: Scan the Keepers (this is really minor though and might not be the most compelling thing to do.)
UNC: Asari Diplomacy
UNC: Besieged Base
UNC: Dead Scientists
UNC: Hostile Takeover
UNC: Major Kyle
UNC: Missing Marines, UNC: Cerberus, and UNC: Hades’ Dogs
Background quest: one of Citadel: Old Friends; Citadel: Old, Unhappy, Far-Off Things; or I Remember Me depending on psychological profile.
Interactions outside of quests:
Barla Von, Volus on the Citadel
Khalisah Al-Jilani, journalist who asks to interview Shepard on the Citadel
Han Olar, Volus on Noveria
For the most part, the shooter parts of the gameplay do improve in later installments, but I wouldn’t say it improves ‘by a lot.’ That aspect is actually worse than I remember and even Mass Effect 2 (and probably 3, haven’t replayed it yet) is a bad shooter. The mission design in ME2 is much better than 1 though—a bit more variety in side missions, some main missions will have gimmicks that help them stand out a bit more (e.g., one planet’s sun burns through your shields, so you want to stay in the shade)—but even so, that aspect is never a strong suit of the series either: “oh look, a bunch of unnaturally arranged crates, wonder if there’s going to be a shootout?.” The cover mechanics are always a bit ass as well. Unfortunately, the game is a cover shooter.
The planet traversal stuff in the vehicle gets removed (aside from an optional series of side quests), so that speeds things up. You say you have bought into the characters, but they also get way more fleshed out in the sequels. I actually found on replaying the first that (Wrex and, to a far lesser extent, Ashley and Kaidan excepted) the way the characters are written, they mostly only exist to exposition dump about their species and do not actually have that many conversations. Which is an okay way to handle introducing a player to the Mass Effect universe, but compared to other RPG companions, they do feel a bit hollow at that point.