I never finished morrowind. (Damn you, Lorkhan. Why does your freaking heart have these particle effects that kill my graphics card?!

)
But judging by what I do in the other few games I've ever finished, I would probably head out to Seyda Neen and have a drink at the Tradehouse, go pearl fishing and generally bring up old memories. After that, I would probably start doing crazy things like levitating to the roof of Vivec's palace and demanding that the people worship me. Finally, I would turn Dagoth Ur's hideout into my new base and begin my own reign of terror.
When the memories and crazy things get boring, I always look for something else to play. In Morrowind's case, buying Tribunal and Bloodmoon. Ah, a person can only dream. Yup, that's what I would do
if I ever managed to get past the flashy heart of Lorkhan.