The Zenith Remnant contains a lot of superweapons, for both you and the AI. Many of these AI superweapons can be captured by taking the entire planet. I'd probably refuse to play the game without The Zenith Remnant.
Children of Neinzul doesn't contain much Fun that can't be turned off in the lobby. That said it's always nice to have some variety in your Fun and I usually have all the Neinzul minor factions set to 1/10 - the Neninzul minor factions in particular are great at adding variety to a game without being too painful. There's also hybrids. They are easily manageable, but a campaign with hybrids is a slower, more defensive game.
The Champion is the main feature in Ancient Shadows. Ignoring champions the feature list is fairly small. The Tackle Drone Launcher and the Wrath Lance are two of the most !!fun!! things in the game, but there's a lobby toggle specifically for the former and you might not ever actually see the latter because AI Homeworld.
Vengeance of the Machine is focused almost entirely on !!fun!! Might be best to skip for your first game, save some surprises for later once you're familiar with it.
I'd definitely set aside Light of the Spire for later as well.
Destroyer of Worlds, sure why not. Once again set aside the Exodian Blade campaign for later and destroy the AI homeworlds the normal way. Ignoring Exodian Blade this doesn't add much, but what it does add includes some hilarious stuff. In particular, the Cowardly and the Mime AI types both sound very entertaining to go up against and I'd love to play a game against them. The preemption plot adds a degree of low-level chaos that a lot of people will enjoy.
TL;DL
Enable Zenith Remanent, Children of Neinzul, Ancient Shadows, Destroyer of Worlds
Disable Light of the Spire and Vengeance of the Machine.
