I probably wouldn't do Lafayette or the Franconia Ridge as a first winter hike. There have been a few deaths up there recently from inexperienced/ill prepared hikers...

Look into Mount Pierce or Mount Jackson, both 4kers with nice views, both get above treeline but not excessive. They are solid first winter mountain hikes, as you can duck back into the shelter of treeline relatively easily if needed (if you didn't quite dial your gear right, or the weather turns).

You might also want to head over to r/wmnf for good tips.

As for breaking into mountaineering, agree with poster who suggested a guide or taking a mountaineering course.