Skills

[Omega Supreme]

What skills should I buy for fast stats and achievements?

[Lucent Beam]

I always made sure I'd buy the skill so that you find better weapons and so that you are more likely to earn dropped skills from enemies.
Honestly though, skills don't matter all that much except for the character you want to do Survival mode with. For that character, make sure you buy Ration to increase your health powerups... but really, by the time you level up that character, you can just be saving all your money and buy all the skills for that one.
Really after I played 2 or 3 Musou modes, I stopped spending my money on buying skills for everyone and just saved it for one character. You really can get by and get through each story mode without buying skills for everyone.