After 2 revamps, the quest chain in Moria will lead you in a linear path though each area (except Foundation of Stone), from quest hub to quest hub.
[Begin] = Great Delving -> Silvertine Lodes -> Durin's Way -> Water-works -> Zelem-melek -> Redhorn Lodes -> Flaming Deeps -> Nud-Melek = [End]

The Epic chain, on the other hand, has not been retouched. And, will send you to go back and fourth, not directly synching with the quest chain. So, the goal of this is to try to match up the Epic line with the quest chain and reduce as much backtracking as possible. Also, I aim to avoid spoilers where possible. So, I won't be telling you how to do the quests.
*Note 1* I recommend starting Moria at lvl 50, beginning at an earlier level may affect your experience.
*Note 2* I recommend you not use any Reputation items. You will be kin with both Guards and Miners before you leave Moria.

Great Delving
As soon as you enter Moria, pick up [Foreword - Book 2] and grab a stable ride to Dolen-view. Talk to the Dwarf there, run south to the Deep Descent, talk to the Dwarf there, then run north to the Chamber of the Crossroads to finish. And pick up [Book 2, Chapter 1]. Now, you can return Durin's Threshold and begin the quest chain.

By the time the quest chain brings you to Dolen-view, you should be ready to finish the Chapter. Return to the Chamber of the Crossroads and pick up [Book 2, Chapter 2]. Head back into the Great Delving and find the items in the 3 Chambers and complete the Chapter. Pick up [Book 2, Chapter 3] examine the fallen plaque outside Katub-zahar and return to complete the Chapter. Pick up [Book 2, Chapter 4] and resume the quest chain in Dolen-view.

After the quest chain brings you back to Katub-zahar, return to the Chamber of the Crossroads and complete the Chapter. Pick up and complete [Book 2, Chapter 5], then pick up [Book 2, Chapter 6]. Then, finish up any remaining quests in the Great Delving chain and head to Silvertine Lodes.

Silvertine Lodes
Simply follow the quest chain through out the Silvertine Lodes. At its end, it'll send you to the Chamber of the Crossroads and turn in the Chapter.

Durin's way
Pick up [Book 2, Chapter 7] and complete the instance. Pick up [Book 3, Chapter 1] and begin and follow the quest chain throughout Durin's way.

After completing the quest chain in Durin's way, it'll send you to 21st Hall momentarily before sending you to the Water-works. However, before you head there, complete the Chapter in 21st Hall and pick up [Book 3, Chapter 2]. The Orcs in Zelem-melek might be bit much for you at the moment, so take a stable ride back to Dolen-View and do the Instance - Training Hall (it will give you credit). Then, return 21st Hall and complete the Chapter. Pick up [Book 3, Chapter 3], run north of 21st Hall and search the rubble. Then, take a stable ride to the Chamber of the Crossroads and complete the Chapter. Pick up [Book 3, Chapter 4]. Head south, back into the Silvertine Lodes, and defend Swiena. Continue south into the Water-works.

Water-works
Simply follow the quest chain throughout the Water-works, and it will send you back towards 21st Hall. Swing by the Chamber of the Crossroads and complete Chapter. Pick up [Book 3, Chapter 5] and head to 21st Hall.

Zelem-Melek
Complete the Chapter and pick up [Book 3, Chapter 6] and its 3 parts. Begin the quest chain in 21st Hall, talking to the Dwarves for the first part of Chapter 6. Follow the quest chain into the Great Hall of Durin and Gazatu-ru, picking up armor, blades and shields for the second part of Chapter 6. Continue the quest chain through western Zelem-melek until you return to 21st Hall.

Now, before you take the goat to Orc-watch, run east out of Zelem-melek to Second Hall in Nud-Melek, stay north and head east until you come to stairs to the north so you can find the junction on Eastern Road. After you find that, you can either head back to 21st Hall, or continue the road north and west to Balin's Camp and 2 optional quests there. After you return to 21st Hall, take the goat to Orc-watch.

Redhorn Lodes
Follow the quest chain, picking up the black powder until you get Auti's Camp.

When you get to Auti's Camp, you'll get a quest to mourn a fallen Dwarf. So, raise a pint and when you wake up in 21st Hall, complete the Chapter. Pick up [Book 3, Chapter 7] and complete the instance. Pick up [Book 3, Chapter 8], head to the Chamber of the Crossroads and complete it. Pick up [Book 3, Chapter 9] and complete the Session Play.

Pick up [Book 4, Chapter 1] and its 3 parts. Hop down the well to the Water-works, return to Durin's Threshold for supplies, and aid the patrol outside the Chamber of the Crossroads. Complete the Chapter and pick up [Book 4, Chapter 2]. Enter the Way of the Smiths from the Silvertine Lodes, and go into the Heart of Fire to complete the Chapter. Pick up [Book 4, Chapter 3]. Head south into the Water-works to kill the Goblin Runners. Return to the Heart of Fire and complete the Chapter. Pick up [Book 4, Chapter 4], return to 21st Hall and complete the Chapter. Pick up [Book 4, Chapter 5].

Finally, raise the last pint in 21st Hall to wake up back at Auti's Camp. Resume the quest chain and it will lead you into the Flaming Deeps.

Flaming Deeps
Simply follow the quest chain throughout the Flaming Deeps, picking up slats and bars, then killing Orcs. At the end of the quest chain end in the Flaming Deeps, it'll send you to Nud-Melek on a goat.

Nud-Melek
As you arrive in Nud-Melek, do *not* advance the quest chain. Instead, return to 21st Hall and complete the Chapter. Pick up [Book 4, Chapter 6]. Head to the Chamber of the Crossroads, hop down the well and find the plaques in the Water-works. Then, head east into the Flaming Deeps and take the same goat to Nud-melek, run back to 21st Hall and complete the Chapter. Pick up [Book 4, Chapter 7] and complete the instance. Pick up [Book 4, Chapter 8], head north to the Peaceful Path in Durin's way and complete the Chapter.

Pick up [Book 5, Chapter 1], return to the stairs of Nud-Melek and begin the quest chain. Upon reaching the bottom of the steps, head south in the Second Hall, heading east. You'll get some optional quests, and it'll lead you to advance the Chapter. With the aid of your Dwarf ally, it'll be easy to kill the Pulpum Orcs, then complete the Chapter. Pick up [Book 5, Chapter 2] and pick up the chests.

Resume the quest chain through the north section of Second Hall, and the rest of Nud-melek, leaving you on the door to Lothlorien. At this point, you're done the linear quest chain. Return to 21st Hall and complete the Chapter. Pick up [Book 5, Chapter 3] and complete the instance. Pick up [Book 5, Chapter 4], return to the Chamber of the Crossroads and complete the Chapter. Pick up [Book 5, Chapter 5], complete the Skirmishes in the Deep-way, the Way of the Smiths and 21st Hall.

Now, you've completed the first 5 Books in Volume II and the quest chain in Moria. You're now ready to head outside into Lothlorien.