I've been playing Morrowind for years but only just started playing Oblivion.

In Morrowind you report to Caius early on and he tells you go off and do some adventuring on your own, join guilds etc. and you can come back to the main quest later.

My question is: does anything like this happen in Oblivion?

I've done several parts of the main quest and so far I seem to keep being told I must do the next step right now or the Empire will be destroyed. Will there ever be a natural point to go off and do my own thing?

I haven′t done the MQ so I don′t know for sure, but I don′t think there really is a natural point for you to go off and do your own thing. Best advice I can give is just to ignore the feeling of emergency and just do what you want to do when you want to. Or maybe someone says something needs to be done right away but you may find it to be a natural point to take a break?

Btw, welcome to the forums and have a fishy stick! --->http://images.uesp.net/c/c4/Fishystick.jpg

Oblivion and Skyrim tried to make the villains feel more threatening, and so added a sense of emergency with no real stopping point. I think there is a "stopping point" near the end

Baurus waited at the boarding house for three years before my character, Baa, went to see him and advanced the main quest. Then it took her another year to finish the MQ. Even doing FG or MG quests, the sense of urgency is just for show. Everything will be there whenever you get around to it, even months and years later.

Unless if you as a PC player use the console to unmark the amulet and toss it in Lake Rumare...