Wondering, what would be the best DLC to rush into after leaving Goodsprings?

I'd probably say Honest Hearts as the "easiest" DLC, but because it's entrance is to the north of Vegas, you'll probably gain a few levels in the journey from Goodsprings to Northern Passage.

However, is there any specific reason you want to do one of the DLCs early? If there's not, I'd recommend waiting for a higher level to do the DLCs.