How does one train for making landings and takeoffs in bad weather?
Your question has two components: 1) how do pilots train for adverse conditions and 2) how do pilots stay fresh on adverse conditions?

As you said, smaller airplanes (really, slower airplanes) are hardest, because the wind speed is a larger percentage of the airplane's flying speed. There are no real simulators for small airplanes: we learn by going out and doing. When you first learn to fly, your instructor makes sure you can handle crosswinds and gusts. With the Instrument Rating comes more adverse weather.

Once you have learned the techniques (and stay proficient by practicing regularly), faster airplanes and higher winds are not really a problem -- the same techniques apply, and in any case you will have a more experienced pilot or instructor helping you along.

I must call out your use of "newbie pilot" -- I hope you're not implying that airline pilots are newbies in any sense of the word. Before you can even hope to get an airline job you've already experienced bad weather in marginal airplanes.

Now, staying proficient: this is where transport airplanes (airlines) are very different from the small airplanes. The airlines' pilots practice regularly on simulators that really can duplicate just about any kind of weather you'd fly in. Pilots of small airplanes though... the simulators can't do that, and we practice in the real world.

All that said, a passenger's idea of what makes a good landing is very different from a pilot's! A passenger thinks a smooth touchdown is what makes a good landing. For a pilot, a good landing means a stable and controlled approach followed by an appropriate touchdown that maximizes safety -- in gusty crosswinds a firm touchdown is much safer than a greaser.