What ever you plan on doing from the start of the character, things such as doing the MQ, Guilds, side-quests, role-playing, or even just doing whatever appeals to you on the spot. Just anything really that you plan on "playing through" with your character.
And as for your question on what to do after you beat the MQ, well in TES games you have extra quest-lines within guilds/factions, a ton of side-quests, a bunch of role-playing opportunities, mods (if you are a pc user, highly recommended
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), etc. You'll find in a game world like this, you will have plenty to do with multiple characters without getting bored.