In my opinion, people should start by increasing their power via all possible means available to them:

refining
purifying
magatamas
Battle armour
etc.

For refining and Battle armor always wait for the monthly event to mazimise efficiency and value.

Then depending on how you want to play, which ninjas you got and what your goal is and your luck you will naturally gravitate to one source more than another. In more detail:

If you want to increase power rapidly for, say the brothers TI then just evenly spread all sources of Power that have already been mentionned before and farm ninjas in order to get them to 5*.

If you want power to dish damage to 9 tails or jinchuriki event, then maybe focus more on magatamas and charms (i think they increased ninjutsu, etc stats) as well as ninja tools.

If you want power for pvp (the most common scenario) then you want to focus completely in initiative. Power doesn't mean much, a higher initiative team with 10k power less than you can still kick your but if its a meta lineup.

For this last option, if you are lucky with cave exploration and have lots of initiative runes, then maybe focus on charms more than on purifying equipment. On the other hand, if you purified a golden initiative stat then focus on getting that to lvl 8 refinement (lvl 9is probably our of the question for even casual spenders). Use your wish tree to get cave keys (2 keys for 15 coupons is good value).

Don't spend on magatamas since you can farm a lot of them and get many lvl 4 on the multiple events and they wont increase your power by that much.

If you want power for pvp go full initiative and when you get lucky with one method build on that. Then repeat and go form there. If you just want power numbers to increase quickly do everything at the same time and make sure you invest fully in group skills at a high lvl group (or convince everyone to donate a lot to increase your group's lvl).

Which method has a better average value/coupon or time ill leave that up to others to debate.