How to use an adjective and multiple nouns with articles?

Flater is correct. The best (clearest and least ambiguous) option is

I have a new note and pen.

Here are some thoughts on these:

I have a new note and a pen.

The repeated article "a" indicates that these are separate items and, because new is located between note and its article, it would be assumed that new applies only to that item. (By eliminating the second article before pen, then the first a, as well as the adjective that follows it, applies to both items in the list.

I have a new note and a new pen.

This is grammatically correct and completely unambiguous. It is also a little unwieldy.

I have a note and a pen which are new.

Again, grammatically correct, but is not what a native speaker would say. When we say "a new pen", it's sort of a shorthand for "a pen which is new". Fluent English speakers understand that adjectives work that way so they wouldn't bother with the longer, more formal sounding structure.