Here’s what happened:

After teleporting Red Skull to Vormir, the Tesseract melted through the floor of the plane and into the ocean. It was found a bit later by Howard Stark, who retrieved it using an underwater drone and returned it to the newly formed SHIELD for study. After the Battle of New York, Thor took the Tesseract to Asgard for safekeeping.

But it was on Asgard during the events of the first Thor movie- I remember Loki touching it, which was part of how he learned he was actually a giant. Thor takes place about a year before The Avengers.

I'll let someone else sort this out lol.

Umm that wasn’t the tesseract lol @Origami Yoda Nerd5

@IamDPorDerpy I thought it was, no? Though I will admit it's been at least a year, nearly two, since I last saw any of the Thor movies.

@IamDPorDerpy Oh, my bad, oops