I've played Morrowind and Oblivion so far. But I haven't noticed anything substantial surviving between the games in terms of story or event. Are there things that have survived from game to game?

In TES4, you famously "closed the gates of Oblivion". So for example, maybe a secret society, inspired by the legend, has since formed. And now (in the next game), they are dedicated to learning every intricacy of the special types of magic and whatnot the creatures of Oblivion possessed.

Do similar things already exist in Oblivion/Skyrim? Or do you have a good idea for such a thing? Feel free to post.

Edit:
On second thought, can't say the idea of bringing lots of stuff in from previous games is a good one. But rather, what is a good idea, is having things in the game that give the player a sense of time in terms of history of the province and of the world. One excellent example is the whole thing with The Blades' fortress in the north of Cyrodil. Truly epic. Another would be dwarven ruins (which I've heard Skyrim is bringing back, so I'm excited).

I think a wonderful thing about the Elder Scrolls series is that you can play any game in the series without missing out on much of the story. Every game acts as its own story and gives you enough information to go on.

That said, there's a lot of continuity that people don't see unless they're really paying attention. You played Morrowind and Oblivion, yet you didn't notice that the events of Oblivion are directly related to the events of Morrowind. Had the Heart of Lorkhan not been freed and had Red Mountain not gone down, then the walls between Mundus and Oblivion would probably have been strong enough to keep the Daedra out, even after the Emperor died and the Dragonfires were unlit. But the Mythic Dawn chose that specific time to act, as they knew the jewel of Red Mountain was gone, that tower was down, so all they had to do was take down the tower in Cyrodiil, the Dragonfires, and they'd be good to go. Oblivion even had continuity from earlier titles. That Uriel Septim you speak to at the beginning is the same Uriel Septim that gets banished to another dimension by Jagar Tharn in Arena. He even makes mention of it, as it's stated that the "other dimension" was Dagon's Deadlands in Oblivion.

By studying Alduin's Wall, we can see that plenty from the past games is now part of the prophecy leading up to Alduin eating the world. We can see the Numidium and we can see Oblivion Gates, so there's definitely continuity.

Well, it'll be moot for Skyrim because it takes place so long after, but I think Oblivion's little bits about Morrowind, where either way the Thieve's Guild seems to have vanished and both things are technically possible, either the Fighter's Guild stamped them out or they went farther underground, and the player can descide which one, for their part, is true.

As mentioned lots of continuity between games. Your story as Nevarine or CoC etc can only be mentioned in the vaguest terms because no 2 main characters are completely the same.