I bought FONV and installed all of the DLC at the same time, so I have the option of playing the DLC without having completed the main quest, or indeed any side quests too. I'm concerned it may mess things up, though. Will I screw with the story line, characters, development of history etc. if I play e.g. Honest Hearts without having completed the base game?

You cant play DLCs after beating the base game.

And DLCs have no affect on the main game.

Ok, so I can just go nuts and go to Zion even though I haven't even killed Benny yet, and pick it back up when I get back from Zion?

You can just do stuff, anything you want. It only affects the DLC world and the only DLC that changes the main game world beyond adding the DLC start location is Lonesome Roads. Even then, it doesn't change much.

Awesome! I thought I had to finish the main quest/base game first, THEN go through the DLC.

Off to ZION! =D

I personally recommend Honest Hearts, then Dead Money, then Old World Blues, then Lonesome Road to get the full "story" in the most unspoiled fashion possible. While a character in HH mentions a character down the road, the mini-spoilers aren't quite as obvious as the one's Dead Money gives for LR, and OWB gives for DM and LR.