I use quite a few mods and have been eager to try more. I can give my opinion on some of them:
Another Portrait Pack: Cosmetic. Not 100% vanilla style but it adds a lot of variety to prevent duplicate officers and I like a bunch of them as PC portraits.
Audio Plus: Great mod which adds a lot of new music that fits the game and updates some of the vanilla weapon sounds.
Blackrock Drive Yards: Very high quality mod with a large variety of frigates/destroyers that feels like they were built for the player to pilot. Favors mobility and going "all in" with their ships and I find almost all of them very fun to play. For example, there's a fragile, experimental frigate with regenerating armor that sucks in projectiles with your right click then blasts out a bunch of energy missiles. Your F ability also jumps through space and blasts the surrounding location with plasma arcs. If you stick to in-faction weapons it doesn't seem too strong. Cool graphics, sound fx, and mechanics.
Combat Chatter: Adds a bunch of messages to the top left of your screen that are mostly helpful. Will let you know when your AI ships overload or get below a certain hull % while there's occasionally flavor text thrown in.
Common Radar: Adds a radar in the bottom right of the screen that I think shows force strength on each side as well as some basic icons. Nice to have though I don't look at it as much as I thought I would.
Console Commands: Used for testing purposes, doing silly things, or performing a "save transfer" if you need to start a new game without losing progress.
Dassault-Mikoyan Engineering: A high quality mod that's not even close to balanced. This is the faction I'm aligned to in my current game and it seems like literally every single ship, weapon, and especially fighter is faster, stronger, longer range, and more efficient than any vanilla (and probably most other mod) equivalents. It's a carrier/fighter focused faction that also adds a hidden boss faction with three spawns of three system clusters in fixed positions NE, W, and S of the core along with I think a couple lone systems here and there. There's a lot of questionable design decisions that are blatantly broken, though the faction itself is fun to play and fight against if you enjoy carrier gameplay. Definitely a pass if you want something vanilla comparable.
Diable Avionics: This was the faction I played with on my first modded playthrough. Extremely cool looking with a lot of phase skimmers and a couple unique mechanics like controllable fighters (that are basically transforming mechs?). Overall it's got very slow, long range projectiles or short range burst beams with a focus on missile saturation and the wanzer mechs. The ships themselves are sleek, mostly integrated with phase skimmers, heavily armed, decently shielded, and somewhat fragile otherwise. The Maelstrom is probably my favorite capital in the game other than the Conquest. I would pass over this faction if you're looking for something strictly vanilla comparable but I had a blast playing it and would say give it a shot if you can look past that.
Disassemble Reassemble: Adds a large number of ships that fit really well into vanilla. Highly recommend.
Graphics Lib: Utility mod that some mods require that makes everything prettier. My PC couldn't handle it but luckily there's an option to turn it off but leave mod compatibility.
Grytpype and Moriarty Defense Authority: I have not interacted with them much though it adds some nice additions to the background and independents. No comment on balance. Doing a non-aligned playthrough next time and will definitely try some of these ships out. Adds a few space cop patrols to hyperspace near their system.
Junk Pirates: If you're looking for a rag tag faction of cobbled together pirate ships you might want to consider this mod. I have not had a chance yet to interact or test their ships yet so no comment on balance.
LazyLib: Utility mod
Lightshow: Adds a greatly needed color differentiation to lasers. Highly recommended.
Luddic Enhancement: This is something that should be in the base game imo. Adds a handful of smaller low tech vanilla ships with new paint jobs to the Luddic Church and 3 civ ships to the Luddic path as well as create a new system with a LC military world and tiny hegemony base. Highly recommended since it adds a tonne of flavor that is much needed to those factions while being as vanilla as you can get.
MagicLib: Utility mod for animations
Mayasuran Navy: I have not had a chance to play or interact with a lot of their ships outside the missions yet so I have no comments on balance. It adds a bunch of neat looking midline ships to the Persean League.
Nexerelin: I can definitely see why not everyone would since it completely changes the way starsector is played. Overall I like the changes and think you should consider it.
Outer Rim Alliance: I think this faction was built specifically with broadside ships in mind and it looks like the AI can actually use them properly. Focuses a lot on high flux cost, long range energy weapons (but not beams) while a lot of ships have a bunch of hybrid slots for missile options or spamming close range energy damage. I have not played around much with them though they seem fairly good at a glance in terms of balance. I'd recommend it.
Planetary Access Shield Control: Makes the planetary shield generator do a bunch of new things with some drawbacks. I think overall a positive change.
Portrait Pack: Adds a bunch of vanilla style portraits to spice the game up.
Practice Targets: Adds 4 different squares that spawn in front of you in the simulator. Highly recommended if you do a lot of ship testing.
SCY: Have not had a chance to play or interact much with these guys yet unfortunately. It has the most in depth lore out of any mod I've seen with two PDFs you can download though.
Shadowyards: Another good faction mod. Adds a few pirate hulls you might enjoy too. Recommended.
Ship and Weapon Pack: Seems to integrate perfectly into vanilla and adds an optional quest chain for super ship bounties that you can ignore. Highly recommended.
Speed Up: Hold middle mouse to increase game speed in combat by 2x. I don't think I can play without this. Recommended.
Sylphon RND: Have not had a chance to play or interact with them as much as I would like but it seems to add a lot of unique mechanics and a boatload of new ships. Low manpower requirements and integration of AI cores seems really cool. Has glowing purple and red graphics and spooky phase jumps. No comment on balance.
Toggle Weapons Groups: I THINK this allows you to swap between alternate and linked fire in combat. Also adds the 6 and 7 keys to control groups.
Trailer Moments: Adds some neat graphics and particle effects to Gauss cannons and such.
Underworld: If you're playing pirates then download this. Highly recommended.
Unknown Skies: Adds a bunch of new conditions and planet types. It really spruces up exploration
Version Checker: Lets you know if there's a new version of a mod at the start of a new game.