Learn Russian: Я читаю этот новый журнал.

The dictionary form is читать (to read). Because it ends in -ать, it belongs to the first conjugation group. To make the "I" form (first-person singular), we drop the -ть and add -ю, giving us читаю.

The English translation says "I am reading". Is there a separate Russian verb form for "I read" generally?

No, Russian doesn't have a separate continuous tense like English does. The single present tense form читаю handles both "I am reading" (right now) and "I read" (regularly). The context tells you which one it is!

Since "this new magazine" is the direct object being read, why didn't the endings on этот новый журнал change?

They are in the Accusative case, which is used for direct objects! However, журнал is a masculine inanimate noun (an object, not a living person or animal). For masculine inanimate nouns, the Accusative case looks exactly identical to the dictionary form. The matching words describing it (этот, новый) also stay exactly the same.

Is журнал a false friend, or does it mean "journal"?

They sound similar and share a history, but журнал almost always translates to "magazine" in everyday Russian. If you want to talk about a personal journal or diary, Russians use a completely different word: дневник.