In the first movie, and some scenes of subsequent movies, Harry puts it on like a regular cloak (that is to say, he wraps it around himself and puts on a hood for his head). This is also the sense that I often got from reading the books. However, when the camera shows someone wearing it later, it is like a blanket over the wearer(s) and the camera, not like a cloak being worn. Is the cloak supposed to be worn like a cloak or a blanket? And, incidentally, how do people see out of it when there are more than one person under it if the inside is opaque?
The first time Harry puts on the cloak it is described as follows:
Harry threw the cloak around his shoulders and Ron gave a yell.
"It is! Look down!"
Harry looked down at his feet, but they were gone. He dashed to the mirror. Sure enough, his reflection looked back at him, just his head suspended in midair, his body completely invisible. He pulled the cloak over his head and his reflection vanished completely.
Though he first puts it on his body and then on his head, which would seem consistent with a regular cloak with a hood, the phrase “pulled the cloak over his head” actually seems to imply that it is one piece of blanket-like material that just needed to be shifted in order to cover his head.
Indeed, there are several other passages that imply your second option as well. For instance, the books often describe a person as being “under” or “beneath” the Invisibility Cloak, such as in Chapter Twenty-Five of Goblet of Fire:
It was awkward moving under the cloak tonight, because Harry had the heavy egg under one arm and the map held in front of his nose with the other.
This would be a very precise description if the cloak is draped over the person’s entire body; a regular cloak with a hood, by contrast, would more accurately be described as “wearing”.
Later in that same chapter we have the following passage:
He lurched forward to try and catch it, but too late; the egg fell down the long staircase with a bang as loud as a bass drum on every step — the Invisibility Cloak slipped — Harry snatched at it, and the Marauder’s Map fluttered out of his hand and slid down six stairs, where, sunk in the step to above his knee, he couldn’t reach it.
Again, this fits much better if we assume like your second option that he was wearing it like a blanket. In that case, the cloak could easily slip and fall off. However, if it was worn like a regular cloak with a hood it would be much harder, if not impossible, for it to slip off.
On the other hand, in the very next chapter we find the opposite implication:
Harry opened his eyes. He was still in the library; the Invisibility Cloak had slipped off his head as he’d slept, and the side of his face was stuck to the pages of Where There’s a Wand, There’s a Way.
If it was worn like a regular cloak with a hood it would be perfectly understandable that his head became uncovered. If it was worn like a blanket, however, it would be hard to explain how the cloak slipped off his head without slipping off his body. We would probably have to assume that instead of having the cloak draped over and around his entire body, Harry instead merely had the cloak spread out on top of him as he was sitting at the table, and the cloak shifted horizontally which exposed his head.
In the first book, Harry is 11. The average height for a UK Male, aged 11, is about 4'8" - however, Harry is described as "skinny, and small for his age". Let's assume that he is instead 4'6".
By age 21, the average height for a UK Male is 5'10". If we assume that the Cloak is designed to be ankle-length for when properly worn by someone of this height (since the Cloak was crafted for the adult Ignotus Peverell), and then subtract the head from each height (approximately a 1') we get a Cloak length of about 5'10" on a body of about 3'6", with a hood on top.
This means that almost 30% of the length of the cloak would be pooled around Harry's feet as a trip-hazard. This fine for standing still (like in the Dorm after opening his present), but a tad inconvenient for sneaking around. Instead, by wearing it like a blanket, it is both less awkward and possible to fit multiple people under the cloak.
There's a nice POV/reverse-POV shot in Harry Potter and the Philosopher's Stone that shows what the cloak looks like from the inside. In short, it's mostly transparent.
Harry generally wears it by putting it over his head like a blanket.