Language?

When class first started I thought "this should be pretty easy, just some symbols and stick them together" but it started going downhill after the Kleene star since I was confused how the * is applied to the language when previously it applied to sigma, which is a set of symbols and how the * applied to the language gives the same result as * applied to the set of symbols in the language.

So I was confused for awhile until the examples came which I surprisingly got them somewhat correct since I did not understand the notations at all. Where did the (1 + 0)* come from when the Kleene star was either applied to the set of characters or the set of all strings in L? (1 + 0) does not use { } so how is it a set? So I kind of guessed that 1 + 0 = {1, 0} because of the + symbol meant union for languages although it was never explicitly said that it applied to symbols as well. All that was said was that (R + S) wasn't really defined, like how saying 1 + 3 is a valid arithmatic expression even if I did not know what 1 + 3 means. To me it would make somewhat more sense if it was ({1} + {0})* instead since (1 + 0)* could be taken as ("10")* for a new learner.

The difference between a regular expression and language is confusing as well so I hope I can differentiate them as I am type this. So according to my understanding, a language is the set of all strings formed by a regular expression. For example, L((10)*) would be the set of all strings {empty string, 10, 1010, ...}. A regular expression on the other hand is not the set of strings but identifies the set of strings. Ok, so maybe that is why (1 + 0)* is used instead because ({1}, {0})* would be a language instead of a regular expression?

Then there's the proof that was never completed in class but whose outline was given. I would prefer the proof (perhaps next class?) since I want to know the proper terminologies to use in this type of proofs.