die Klasse is a bit more general and refers to a grouping or classification of things like social class or something like "the class of vertebrates" in biology. In the context of schools it usually means a group of students in a grade. "Unser Sohn ist in der achten Klasse." -> "Our son is in the eigth grade."
der Unterricht (masculine) is the actual lesson/instruction/teaching itself. "Wir sollen ins Kino gehen! Heute hat mein Unterricht keine Hausaufgaben!" -> "We should go to the movies! I don't have any homework for my class today!"
Hope this helps! Not a native speaker, so any corrections or clarifications are appreciated.
"Wie" means "how." Casually, in English, "How is X?" can also be phrased as "What is X like?" Like, here, is an adjective rather than a verb: "It's like drinking from a fire hose: there's too much!"
So "How is your new class?" can also be phrased as "What is your new class like?"