It's really hard to assign a perfect order, but it's clear that you should get everything but ARM and maybe GSM to 15 pretty early on. ARM gets Rapid, which is kinda blah, and GSM gets Manipulation, which is just a more efficient version of Master's Mend I. Otherwise, though...
CRP 15 -- Rumination. No drawbacks, awesome skill. You want ASAP.
WVR 15 -- Careful Synthesis. Almost no drawback unless you're in that gap between 90% not completing a synth and 100% completing the synth. Useful especially when leveling.
CUL 15 -- Hasty Touch. Combine with Inner Quiet to push your Control to astronomical levels.
LTW 15 -- Waste Not. Pretty meh at endgame, but excellent while leveling. Generally gets you 15 dura for 56 CP, which is two full synths unless you tried to hit the ability twice. Cheap enough that you can afford it with earlier gear.
Eventually, you'll also need
BSM 15 -- Ingenuity. Almost essential for endgame crafting. Feel free to skip until then.
ALC 15 -- Tricks of the Trade. OK while leveling, but especially shines at endgame.
As you get later, you'll really want
CRP 50 -- Byregot's Blessing. Hasty/manipulation/tricks spam for building control, pop Byregot on Good or Excellent. Get it before SE nerfs it into the ground and HQ gets harder.
CUL 37 -- Steady Hands II. Very useful for obvious reasons.
GSM 50 -- Innovation. Again, endgame skill for increasing chances of that final 100% HQ pop. Good skill. Helps conserve dura later when you're aiming to pop Byregot.
Also kinda nice to have on occasion, but nonessential:
CUL 50 -- Reclaim. I guess it's OK if you're big on risks. It gives you the chance to go nuts with Hasty and hit the reset button if things don't go your way, but it's not 100% success, and I'm not all that keen on even a 10% chance to lose 150,000 gil worth of Coke. Would be great for making one-stars if there were a market for them. But alas....
WVR 50 -- Careful Synthesis II. Good, but not essential.
ALC 50 -- Comfort Zone. Small CP regen ability. Good to have. Wastes a step with an effective CP gain over time, giving you more time for Tricks procs and saving dura for Control boosts, but takes quite a few steps before you're back in the black. Definitely will have a place in your rotation, but not essential to get.
BSM 50 -- Ingenuity II. Replaces Ingenuity for the most part, but the benefit isn't large by comparison. Again, good to have, but not essential.
Skills you can skip out on for the most part:
-All level 37 skills except CUL, which are Brands. These are only good for a few recipes while leveling. I think there is only one cross-class recipe per crafter, around level 37 or so (some Vintage this-or-that).
-ARM 15 -- Rapid Synthesis is only really useful if you've screwed up and need a last-ditch swing to have a hope of finishing it at all. I've done so before (needed ~210% progress at 30 dura, failed Standard, so I had two whacks of Rapid or guaranteed fail), but I don't recommend planning on it.
-ARM 50 -- Piece by Piece is crap. Ingenuity -> Careful will give you more progress than Piece by Piece, plus the Ingenuity for the next 5 steps. Can never finish with Piece by Piece. Having it on your bar is a waste.
-LTW 50 -- Waste Not II. Basically never use it. Costs more than Manipulation, and if you use two non-dura moves over the 8-step interval, then you're losing out by comparison.
Therefore, if I were to make a plan: skip out on ARM altogether; raise LTW to 15; make sure you have 15 BSM, CUL, ALC, WVR; max CRP ASAP; then do everything else you feel like doing. Don't do ARM unless you really want to make armors (and there is precious little room to make armors right now, with Coke the way it is). Only do LTW if you want to make money, but it's very good at that.