The majority of the city is blocked off for now, so your route to the Lhusu Mines is fairly linear. Feel free to interact with the locals and shop, and don't forget to buy maps off the Moogles of the Cartographer's Guild. Save at the crystal, then head into the mines.
It's a good idea to use Libra, because there are traps everywhere, but they show up as pillars of light if you have Libra enabled.
When you enter the mines, you'll see a short scene. Be sure to explore the mines fully, though for now there's only one path you can take. Keep an eye out for traps, because they hurt! When you reach the Shunia Twinspan, be very careful crossing it. The skeletons here will continuously spawn, and while one or two may not pose a threat, more than that do. Do your best to defeat them one at a time. Alternately, you can take the path to your right and avoid the bridge entirely, though you'll miss out on all the LP. There is a chest down there, but it's booby trapped. If you approach it cautiously, inching along until the !! appears above your head, you can open the chest without triggering the trap.
Continue on until you reach a large room where you'll see a cutscene involving Lamont. Suddenly, Ba'Gamnan and his cronies burst in! If you're fast, steal a Round Shield from Ba'Gamnan, then take Balthier's advice and run! While this fight is winnable, you have to be around a level 14 or so, and unless you've done some serious grinding, you're more likely around level 10. They'll most likely pound you into the dirt, so hold R2 and flee all the way back to the exit.
You'll see another scene, and it's decided that Vaan will run around town like a madman, proclaiming to be Captain Basch fon Ronsenburg. Only people with talk icons above their heads will respond, and the majority aren't worth talking to. Your best bet is to seek out the Parijanah, who stand around town with books as a sort of tour guide. They'll increase your notoriety quite a bit, but watch out for the Informed Sainikah, however, because they will decrease it just as much.
Once your meter hits 100%, you'll watch a scene that takes place in the tavern. Once it's over, check the Notice Board for the Nidhogg Mark. You're probably still too squishy to beat it, but at least that step is done. Once you're ready, go meet with the Marquis to further the plot.