Should I explore every dungeon and structure as I encounter them in my travels or will a quest eventually lead me to all locations?

No, quests will not bring you to every cave or other area on the map. Skyrim was made to be a game where not everything was bound to a quest, although they are fun. It is worth it however to search these caves because they will usually contain decent loot and you will level skills from fighting enemies. Have fun!

Yes and no:

Thanks to the Radiant Quest system, pretty much every location in the game can potentially be the target of a quest; but the odds of this happening for any given play-through are slim to none.

If you happen on a cave or fortress, but don't have a quest there yet, exploring it is usually worth the effort; but keep in mind that you might get sent back there for a quest at a later date (at which point the place will reset, forcing you to clear it again).

Some quests tied to dungeons are unmarked, which helps make the game feel less gamey; and may also prevent that dungeons from being a target of radiant quests; so even if you spam a bartender or other quest giver for repeatables in any given region, you'll still not get sent everywhere.

Finally, it's worth noting that some caves have bugs related to resetting, and quest completion order; potentially preventing you from completing a quest if you clear the dungeon before getting the buggy quest.

However, if you only do the main quest and resign yourself to free-form exploring the rest of Skyrim, this is unlikely to present an issue.

Quests will not always lead you to one but there are some that do. However the ones that you do find you should walk up to them until the name of the location pops up so you can fast travel to it later.

A simple answer to your question, no, quests won't lead you to all of the dungeons so make sure you explore A LOT!