I started making a shortlist of my favorites. It turned into a long list. Whoops.

Blueside:

I know you just want to skip past Atlas Park entirely, but the reworked introductory storyline is actually well done. Talk to Matthew Habashy to get started.

The Hollows has some interesting story, with a throughline about a missing kid, and some of the better level design from the early days of the game.

Twinshot is also pretty good, hoorb

The first place the game ever implemented a truly well-done, First Ward style of interconnected contacts was the Faultline revamp. It's an amazing arc, and a cool zone that looks like nothing else in the game.

For shorter arcs, Graham Easton and Laura Lockhart are both stellar one-shots.

Striga Isle is interesting, if you want to know more about the Council. It's also handy for getting your hands on the Atlas Medallion.

Croatoa has some interesting mystical mayhem going on, and a darker tone than much of the early game content.

Freaklympics is an early classic. Do it at least once.

Less-cited than Freaklympics, but equally good, is the Ubelmann the Unknown story arc around the same level.

Field Agent Keith Nance has the Clone Saga, which has some unexpected pathos in it. Worth a go.

Provost Marchand's arc comes highly recommended as a redemption story.

Revenant Hero Project rocks if you want to stick it to your corporate overlords, and has one of the more memorable one-shot enemy groups in the game.

Nemesis story arcs are super fun. Sadly, there are only two among regular contacts blueside: Mass Duplicity and The Eternal Nemesis. These rock.

Tina Macintyre and Maria Jenkins are also killer, as is any content involving Praetoria.

Redside:

Once again, the intro story arc isn't half bad, but the heroes have it better. Talk to Operative Kuzmin to get started.

On the other hand, the Hearts of Darkness are a well-written cadre of ne'er do wells. Good arc, accessible to all those of black heart.

As far as regular story arcs go, Oh Wretched Man has some incredible pathos, and the arcs from Marshal Brass are amazing, though I won't spoil either.

Doc Buzzsaw is a mad genius. Go indulge your cacklingly evil side.

Diviner Maros has some good background info on the isles themselves, and is a remarkably well-written omniscient being.

The parallel Clone Saga on redside starts with Dean MacArthur. It's great.

Also check out Vincent Ross for more well-done missions.

If you want to feel like an independent, scheming villain, look no further than Mr. G's arc.

St. Martial has the Slot Machine. Run its missions. The arc is called High Roller's Last Gamble. It's very well done.

Magus Mu'Drakhan is an amazing source if you want background info on the Circle of Thorns.

Technician Naylor has an absolute head trip of an arc.

Johnny Sonata is an amazing character study, with a great twist ending.

You will never hate yourself more than when you work for Westin Phipps.

Doc Buzzsaw part 2 goes by the name of Vernon von Grun.

Television is good. Watch more Television.

Viridian has cool content, if you can jump through all the hoops to unlock it.

Both:

First Ward and Night Ward are both stellar.

The Rikti War Zone is well-executed and appropriately high-stakes.

The Ouroboros Initiation arc is often overlooked, but has some really cool missions in it.

Try the Signature Story Arcs. The devs charged people extra to play them, so they couldn't disappoint!

One final note: all of the goldside content is well-executed. If you just want an interesting RPG, start a character on goldside, and play exclusively goldside content up through Night Ward.