While the Statute of Secrecy was not mentioned explicitly in book 1, it was mentioned many times that the wizarding world is a secret. Just to name two instances:

• Chapter 1: McGonagall clearly disapproves of the overblown celebrations of Voldemort’s disappearance (lots of owls, shooting stars), even stating that it would be a real mess if muggles found out about wizards. (No verbatim quote, since I don’t have the english book available right now.)
• Chapter 5: Hagrid explains to Harry that the main task of the Ministy of Magic is to keep the existence of witches and wizards a secret.

(Families of muggleborn witches and wizards will learn about the wizarding world, of course; but other muggles won’t.)

With the rat, it’s less obvious: I would even argue that the relatively frequent mention of Ron’s rat in book 1 is weak evidence for it not being a normal rat. Plus, there’s a scene during the train ride to Hogwarts, where the rat is smashed into a window pane violently: Would a normal rat survive this without any apparent damage? Possible, but rather unlikely, so that’s additional evidence for the rat being somehow magical. (Wizards being more resistant to force than muggles is mentioned several times in the same book, as is the existence of animagi.)

Regarding Azkaban: No mention in book 1, yes, but it is mentioned in book 2, before it started to play a major role in book 3.

Regarding the last point (the cloak being special): Well, it belonged to Harry’s father but he gave it to Dumbledore for savekeeping. That alone is evidence that this is not just a normal cloak with a simple charm on it, which you could just buy again, if you lose it. Ron even says that such an invisibility cloak is extremely rare and valuable.