You need at Cut to complete the story, Fly is always helpful, so is Flash in caves, you need Surf, Waterfall and Strength to get every item and to complete the post-game story, and you need Dive to access one of the most fun things in the game, the Abyssal Ruins.
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If you want to limit it to two, Basculin and Rufflet work for me. As the other guy said, though, aside from one use of Cut near the start of the game, required HM use is postgame-only, so don't worry about it too much. Early on in the postgame (or possibly on Route 10, IDK) you'll get a Super Rod from someone, and if you fish in a "rippling spot" (the water equivalent of a shaking grass patch) then you stand a pretty good chance of catching a Basculin. The one I caught was 20 levels above the rest of my team!
Rufflet's White-exclusive though, so if you have Black, go with whatever the version-exclusive bird is that Black gets. They're both found on Route 10, so if you do a bit of exploring you're bound to find one eventually.
The nice thing is that there's only 6 HMs this time around...
EDIT: You're also going to want something that can learn Flash. I just gave it to Zekrom, but anything that can learn it will work. If you do give it to something you want to use seriously later on, just make sure you either have a spare Heart Scale or don't make any changes that a TM won't reverse.
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