Should I play Oblivion after playing Skyrim?

First of all the stories between Oblivion and Skyrim are different. They are also set hundreds of years apart in two entirely different land masses.

Second, go ahead and play the other games. The whole ES series has many stories to follow up on. Going backwards will be interesting. It will be like a historian going back in time to re-live the past before the events in Skyrim.

If you get the chance, try it like this:

Skyrim > Oblivion > Morrowind > Redguard > Battlespire > Arena > Daggerfall (and their expansions)

There were other ones, but they were made for other platforms.

Oblivion takes place about 200 years before Skyrim, so only general Elder Scrolls lore is shared between the two. The lands you play in are different. However, there are some things you do in Oblivion that are referenced in Skyrim (the final battle in Oblivion, for example), but only in general conversation.

Going backwards won't do much as far as the story goes. Just be warned that it is a bit different from Skyrim. The leveling is different (and harder, some might say), you have to constantly repair your gear, and you can only dual wield swords (not a sword and a spell, for example, or two spells).

If you do get Oblivion, get the game of the year edition if you can, as it has the 4 expansion packs and other various DLC. By far, the Shivering Isles is the most fun of the DLC, imo.

As the games take place in different time periods, it doesn't matter what order you play them in, so going backwards will not be a bad thing.

Just keep in mind the technical and gameplay differences.

Sure. It's a great game, especially if modded. One problem I found with it however is that it is too leveled.

I played Oblivion first and then went backwards and am playing Morrowind. It's a good game, with some flaws.

I don't see why not. It's not the same story.