The highest physical attack value belongs to the fire dragon scale, but grimoire scale's got the best overall stat bonuses. Earth dragon scale increases defense the most, and i can't quite recall what the water dragon one does but it's probably related to magic attack or something.
Most people stick with grimoire scale.
You know you're running out of ideas when one of them is a badly drawn face on a badly drawn purple cloud, attached to what appears to be a coaster.
For weapons, I generally use 5 Grimoire Scales, 1 Chimera Tail, and 3 Legendary Scales. Gimoire Scales are the best for greatly increasing attack value and other stats withough reductions, though its DEF increase is slightly lacking. The Chimera tail also increases a fair amount of power, but it also allows you to poison or paralyze enemies, which is a good way to level up your own resistances. The Legendary Scales are for absorbing HP, which would greatly cut down your need to carry potions and foods for recovery.
For armor I generally use Dragon Fin. But if you can't get those yet, then Electric Crystal. I don't like the Grimoire Scales' weak DEF increase, so I use the Dragon Fins (or the Electric Crystals) to increase DEF value, since they increase DEF well without reductions.
But these are just MY current methods. A lot of people use different weapons and strategies, so I'm sure they have different ways of upgrading their equips to better bring out their own full potentials.
Max damage is not always the best option, in my case i just made an Earnest Edge and fully upgraded it with legendary scale. The damage is not high, but the blade drains about 75~90% of the damage, add an Art of attack for greater range and then a Rail Strike skill and you'll be tearing through the dungeon at lighting speed and using almost no food. The strategy can be done with any weapon and Rail Strike works with anything, even hammers and staves. It also allows you to go full meele with bosses.
Now if you still want great attack, find the best combination around the itens shown, but it's always good to have at least 2 upgrades with legendary scale.