I think the most important part of learning how to sing in different styles is having the basic backbone of good singing technique. When you know how to use your voice in the best way then you can start to experiment with different styles of singing. Strictly learning one style of singing only hinders you in the long run because it shapes you into a box when another style might need you to be a circle. I always focus on good technique with students before we try and sing through different styles of music.
Another way I try and teach and even learn new styles of singing is to listen to other singers. If you want to learn how to sing like a classical singer then you should listen to all different kinds of classical singers. Same goes with pop or jazz or even rock. There are so many wonderful singers out there to learn from!
But the main point I want to get across is, the sound or the style might change but the technique should never. A good singer is a good singer no matter what they’re singing.