The entrance to Frostmane Hold is west of Jessup. You can't miss it; it's a big cave. In vanilla WoW, this cave was the source of infinite frustration. Originally, the respawn rate in the cave was so fast that if you died, you could barely recover before new mobs were attacking you. But in Cataclysm, Frostmane Hold is much more manageable.
Boss Bruggor is at the bottom. He's distinct from his fellow troggs by his large size and red hair. Simply kill him. Near the dead boss, a detonator is waiting for you. Right-click the detonator to blow up the trogg tunnel! Navigate your way back to Jessup. When you report your success, the dwarf tells you that it was a job well done, but there's One More Thing. Jessup asks you to take his report back to Mekkatorque.
Return to New Tinkertown. When you turn in Jessup's report to Mekkatorque, the King of Gnomes warns you that Crushcog's minions have taken over an armory to the north. You'll have to go deal with them shortly. (Get it? Gnome? Shortly?) Grab the quest from Hinkles Fastblast to blow up some of Crushcog's Mechano-Tanks. It's your way of saying, "Thanks, but No Tanks!" Before you head out, take the possessions from What's Left Behind to Tock Sprysprocket. Take the time to watch the in-game action; it's hilarious.
Follow the road north out of New Tinkertown to get to Crushcog's Arsenal. Once there, use the grenade in your inventory to blow up tanks. Kill the minions in whatever way you enjoy doing that kind of thing. Then return to Mekkatorque. As soon as you turn in the quests, you'll get the opportunity to overhear the next stage of deployment against Crushcog. You will be sent to the Staging in Brewnall. Follow the road southeast to get to Brewnall Village.
In Brewnall, you are greeted by Jarvi Shadowstep. Jarvi informs you that Crushcog has taken up a defensive position in the middle of the lake. In order to sneak up on him, you'll have to run out there and paint over the Sentry-Bots. Go out to the lake to the east, and when you see a red bot, you have to Paint it Black. (Use the gun in your backpack.) Turn in this quest back to Jarvi, who will give you Down with Crushcog!
The Fall of Crushcog
Run back out to the center of the lake. Mekkatorque is standing there with Mountaineer Stonegrind. Speak with Mekkatorque to start the event. Be careful not to aggro Crushcog himself; let the NPCs handle him. In the meantime, you should focus on the minions who are attempting to defend the boss. Once you've defeated Cruschog, return to Brewnall Village. (If you happen to die during the event, don't worry -- it's a very short run back from the graveyard.)
When you turn in the quest to defeat Crushcog in Brewnall, there is one more brief celebration. When that's finished up, Jarvi gives you the quest to enter the mainstream quest area. Your gnome is On to Kharanos. Follow the road west to get to Kharanos. You can pick up Bound for Kharanos along the way. Be sure to check out the awesome gnomish border along the way; it's been quite jazzed up for Cataclysm.
Visit the WoW Rookie Guide for links to everything you need to get started as a new player, from how to control your character and camera angles when you're just starting out, to pulling together enough cash for mid-level expenses such as mounts and dual specialization, to what to do when you finally hit level 80.