Need Some Seaonsed Buccs to Weigh In

Barrage before demolition. Definitely with the barrel thing where you can reset your cooldown. KurayamiLove knows more about that one.
Demolition is your strongest skill, but you can only do it when supee transformed, which you won't be half of the time (when your super transformation is maxed). For your maximum demolition potential you'll need max demolition and max super transform, which is 60 sp or 20 levels.
Imo go for the normal route as you won't be bossing before 135 or so anyway. Your dragon strike grinding will be nice.

 

You gain early damage with barrage first but trade for less versatility in build path later on. Havent run any numbers to account for how much is gained with barrel tech at various skill level combinations, but without considering it demo outpaces barrage the moment super transform is maxed (level 140; 11 si, 30 demo, 20 st, and 1 in barr, ds and tl). This allows you to branch out to higher 1v1 damage by getting Barr, better grinding with ds, or insane utility with tl. You really do want max TL not long after HT level (155) at the very latest. You arent gonna do damage at that level, but tl will literally save boss runs (bonus points for better afk zak runs too).

A traditional barr first build has no flexibility in their build path. You'll be practically forced into the si11->barr30->ds30->demo30->st20->tl30 path, not maxing TL until 170, and not even being able to transform and know what the class is all about until you get demo 20 at 150 (by which point youd actually be weaker than a demo first build). Again, idk how much of a dpm boost barrel-cancelling provides compared to rushing demo, but my gut says it shouldnt be enough to overtake the damage, support, and versatility a demo first build can provide.

 

Instead of listening to all the bozos in the world, get to level 120 and try it out yourself! I.e. at level 120, put skill points into:
-Barrage 1
-Super Transform 1
-Demolition 1
Try them all out and see what you like more: Barrage+Double Uppercut/Barrage+Energy Drain combos? Or does Transforming and using demolition seem more fun? (When transform is down, use Flash Fist to charge up Energy Charge and spam Energy Drain)

Note: if you’re going for a bossing build, Dragon Strike only adds marginal damage so I would not recommend it (but it does looks very cool hehe)

Either ways, you’ll first get Time Leap level 1 & Speed Infusion level 11 and then max Barrage 30 if you like barrage or max Demo 30 + STransform 20 if you like zoom zoom attacks. And then max the one you didn’t next. Lastly, max Time Leap to make you indispensable at harder boss runs!