Thing is, I do not like to rush in, but then I find that when I try to plan it out, everything works out for my friend, who certainly does just rush in mindlessly! Irony! But I don't really know any styles that I'm good at that are good against Bowser. Can you give me a few characters? I usually play as Squirtle, Roy, or sometimes Lucario. I just really like their styles.
I will say, planning on doing something ahead of time is a bad idea, and rushing down mindlessly is a bad idea. Having an idea of what your safe options are is not the same thing as committing to said plan prematurely. You want to know your options (and his) and react to anything he commits to as well as take advantage if he lets you get a little too close.
And to give you an idea of what your safe options are, pretend that he does the perfect response to your attack. What happens? If his perfect response would put you in a bad position (e.g. you lose being closer to center stage or you are above him), or if he can hit/grab you to punish, then you didn't attack safely. either space your aerials so you hit when you're landing and at a distance that's outside of his Out of Shield (OoS) options, or hit the shield late and follow up with low start and end lag ground moves like jab combo's and maybe mix in a grab attempt.
Note: unless maybe you're Jiggs or Wario, hitting people in the air when you're not near the ground to L cancel it asap, is probably not safe. Another exception would be multi-hits, only because you could continue to apply pressure until you land (assuming you land around when it ends or before).
Also Bowser sucks at approaching, so if you force him to approach via a projectile, and then you just get ready to react and punish (probably by shield grabbing most of the time), then he'll have a really hard time.
Bowser is only hard in that if you don't take him seriously, even for a brief moment, you can take massive damage or lose a stock. But if you always respect his options, it'll be damn near impossible for him to hit you.
Also Lordling's advice was good. The only thing he got wrong was that the yell is not optional. The yell will crush his spirit, allowing you to take two stocks with one strike.