Rêver really means “to dream”. “Tu rêves” (“You’re out of your mind” in this context) is an idiom so I don’t think the given example is proper to any kind of meaning explanation, since it could be replaced by thousand other things that have no link to the act of dreaming. It simply refers to the act of dreaming, in its most basic conception, and you could translate it by “to dream” in all cases I believe.
Songer is more of a literary term, that actually feels closer to “wonder” than to “think”. It implies some daydreaming. It could also mean dreaming, but I have the feeling that the sense of “dreaming” is kept for the noun (un songe = a dream). It’s also a very literal term that you don’t use in modern French without the will to add some “poetry” to what you’re saying, either to sound formal or polite, or to imply a sort of “amazement”, a sort of general poetic state of mind. In modern French, you wouldn’t use “songer” to speak about nightmares for example, it would sound really ridiculous and pretentious. You’d keep it for sweet dreams, dreams that are worth some poetry. In old French, you could totally use “songer” or “songe” for both sweet dreams and nightmares.
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