In Megaton, you have 3 options to learn where dear old Dad went:
1) Talk to Moriarty who will, for 100 caps, offer to tell you where your father is. If you don't have the caps the price will go up to 300 the next time you speak with him. (He may offer a discount if you agree to kill Silver (in Springvale) and bring back the caps she stole (though you can talk them out of her, without killing her)).
2)If your speech is high enough, you can convince him that your father mentioned him.
3) For a minor hit of bad karma (very small, and it's the route I generally go with), you can A) steal his terminal's password from the locker next to it, B) use the "Lady Killer" perk (if a male character) to seduce it off of Nova (or, off her dead body, if you're into that sorta thing, sicko). Or hell, just hack the damn thing without it (Again, what I did).
Make sure you're well prepared for the next part.
Follow the objective past Super Duper Mart and Wilhelm's Wharf, and cross the river. It's smarter to use the bridge than to go for a swim, just sayin'.
Head for the first Metro station you come across, either Farragut West Metro Station or Friendship Heights. From there, start wandering the tunnels (pay attention to your Pip-Boy map, it will tell you where the exits are if you're somewhat around them). This is where preparation comes into play. There are quite a few Feral Ghouls and Raiders running around. Follow said tunnels until you reach Chevy Chase. (the location, not the actor...that joke sucked)
Follow your objective marker and it'll take you to the GNR building. Follow the soldiers, helping them attack Super Mutants, and watch out for the Super Behemoth when he attacks. (Don't forget to loot everything! Especially the soldier with the Fat Man and the Behemoth)
Afterwards, use the intercom, enter the building, and talk to Three Dog. There ya go!
Also, it's possible to complete this another way if you find Vault 112 before talking to Three Dog (the quest will be marked as completed). Once you make it to the GNR building, you still fight all the mutants (and the big fella), but on your own. Fight them, kill them all, use the intercom, talk to Three Dog, and Bam, done.
Hope this was helpful. :)