Just writing this out mostly for myself, but also for anyone else who might want to run NG+ sometime in the future. These are the all the story nodes you must activate to complete the story, in order from start to finish.

(Any nodes with a >> next to them can be skipped, so I won't note them. Most of the nodes with a Soul Flame (I'll call them "battle nodes") or Story Book ("story nodes") on them must be completed, with a few exceptions.)

Substitute Shinigami Arc

Need all 7 battle nodes, from left to right. I'll just note here that you can't kill the hollow in "Rapid River of Fate I" until Uryu stops yapping, which is annoying

Soul Society Arc

Only need the one battle node at the start, can skip the 3 story nodes surrounding it.

1st Ring

Need all 4 battle nodes. Can do Chad and Uryu's (top and bottom rows) in any order, but must do Ichigo's (middle) from left to right in order. Once you've beat all 4, it unlocks "Like a Flower on the Quay," which you must watch, i.e. skip through.

2nd Ring

Need all 4 battle nodes. Can do Chad's (middle row) first, or after completing Ichigo's (top row) from left to right. Unlocks the story node "A Chance Encounter," which you must watch.

3rd Ring

Must complete the 2 battle nodes on the top row from left to right. Can watch "Pride and Blade" before or after those. YOU DO NOT NEED "FURY," Hitsugaya's fight against Gin at the bottom. Can completely skip it and you will still unlock "Dreams of Penitence," the next story node.

4th Ring

Need all 3 battle nodes and the story node. Just make sure to do the top battle node before the top story node and you'll be good. Unlocks "Sokyoku Unleashed."

5th Ring

Need all battle and story nodes. Can do each row in any order, but you must complete each required node from left to right. For the top row, must complete "Love and Hate are a Poison" before doing the two branching nodes. Unlocks "Tenteikura." After watching that, must complete the 3 story nodes and the 1 battle node afterwards, from left to right.

Visored Arc

Over half the nodes here are skippable, so to make a long story short, you don't need any "Intermission" nodes (the ones with big, circular character shadows). Other than that, just go from left to right. Whenever it branches, complete both sides of the branch from left to right before going on. And of course, you can skip the nodes with >> on them.

Hueco Mundo Arc

This is the arc that made me want to write this down, because the start is a little confusing. Complete the two story nodes and the battle node at the start from left to right. No problem there. From here is where it gets a little complicated:

Red Key

First, complete the middle node of the center 3, which is Ichigo's. This unlocks Renji's node on the bottom.

Blue Keys

Complete Chad and Uryu's battle nodes (top and bottom from Ichigo's), and Renji's story node in any order. After completing all 3, you will unlock Rukia's node, the one that branches off from Chad's node.

Green Key

Now you can complete Rukia's battle node. This unlocks the rest of Ichigo's path (middle row) as well as Renji's path (bottom row).

Grey Keys

Must do all the nodes in Ichigo and Renji's paths from left to right. After completing the node at the end of both paths, you'll unlock the story node "Rescue." Watch it, and things will get a lot simpler.

2nd Ring

Complete all 3 battle nodes here in any order. This will unlock the two story nodes to the right before the next ring.

Purple Keys

Need to do the 1 battle node on the bottom path, and the 2 battle/story nodes on the top path first. This unlocks the next node in each path.

White Keys

Now just watch the story node on the bottom path and do the 2 battle nodes on the top path from left to right. This unlocks "Site of the Final Battle," the final story node of this arc.

Battle for Karakura Arc

This arc is simple, thankfully. You need all nodes that don't have a >> on them, from left to right. Complete all the nodes in each row of the branches from left to right before moving on, you probably get the idea by now. The run ends after beating the last battle node on the far right, when the HUD fades after using Mugetsu on Aizen.