From a burst attack perspective, you want the highest rarity possible as your main weapon. An SR with no * will still have a higher multiplier than an R with *. Likewise, SSR with no * will still have a higher multiplier than SR with *.

The relevant formula's basically (normal attack * (a + burst_up_effects) + b).
For the heroic soul...
For R, a = 2 and b = 1500
For R*, a = 2.5 and b = 1500
For SR, a = 3 and b = 2000
For SR*, a = 3.5 and b = 2000
For SSR, a = 4 and b = 2500
For SSR*, b = 4.5 and b= 2500

For kamihime, a diverges at SSR
For SSR, a = 4.5
For SSR*, a = 5
For... post-awakened SSR I guess? a = 5.5 and b = 3000

By the way, keep in the back of your mind that the multiplier's written as (a + burst_up_effects) to understand how impactful burst damage buffs are. It's especially relevant when Shingen is eventually released and you see that 'up to +200% burst damage' buff.

-

As far as weapon types go... IIRC, a character gets +20% stats from a weapon if it's the preferred type. With the heroic soul, it's easy to see: it's the weapon(s) it can use. For kamihime on the other hand, their preferred weapon type isn't listed in game. You either have to pay real close attention to see who prefers what, or consult the japanese wiki because the users already did that work.

It's a mechanic that you should keep in the back of your head as an explanation for some things, but don't stress yourself over this. Focus on the things the game already presents to you; the stats and assault skills. Factoring in weapon types to figure out how to squeeze every last drop out is a late game complication and it can impact your party compositions.