Treasured tools is a good way

Cave keys: rebates gives you rune stones and also treasured tool essences. Make sure you do full rebate (1050 or 2100 keys)

Refine runes: also make sure to save up for a rebate to make full use of wish credits and additional rewards

Other than training pots (which have great value to gain power) those are the main two things you should spend you moon and sun scrolls.

Your treasured tool will give you great ini, put all your essences in one tool and give it to your P1 ninja.

I never spend on purification runes, you'll get enough from events, so make sure to purifiy your P1 ninja with gold ini stats.

You'll get additional ini from things like summon cultivation and myoboku trainings, but I don't spend moon scrolls on those.

All the things that gives Initiative in the game:

1. Cave Keys
2. Training Potions
3. Assist links
4. Secret Scrolls
5. Myoboku
6. Battle Armor
7. Having Ninjas
8. Treasure Tools
9. Refines
10. Charms
11. Five Nature Chakra
12. Summon Cultivation
13. Quick rune, in Eigth Inner Gates
14. Purifies
15. Home Bonds

But as of right now, the best way to get Initiative, is through Training Potions.
Spending Moon Coins on Cave Keys, and spending your Sun Coins on Advance Refines, or Epic Refines.
Depending on what you need it the best way to utilize them.