Despite the way Wizards of the Coast has positioned them, I don’t think that Waterdeep: Dragon Heist and Waterdeep: Dungeon of the Mad Mage really pair together very well. One is an urban intrigue and investigation campaign, and the other is a chthonic megadungeon. These have very different tones and structures, and many players will strongly prefer one type over the other. Even if they love both types, intrigue and dungeon delving don’t mix together well: oil and water, not peanut butter and chocolate.
Folks interested in a dungeon-focused campaign have other options, though. The introduction to Tales from the Yawning Portal provides just enough narrative framing that you could use it as hooks for a group of entry-level adventurers and send them off to the Sunless Citadel and Forge of Fury. Alternately, since DotMM starts at about level 5, it could follow up on Lost Mines of Phandelver (the Starter Set adventure) pretty nicely.
Alternately, go take a look at Adventure Lookup (organized by Matt Colville but largely a community project) and find something that fits your group. Or, you know, build your own. Half the fun of running a megadungeon is designing it yourself, whether drawing the map or grabbing someone else’s and stocking it with monsters and traps and treasure as you see fit.
Most of my friends have already played through those first couple of dungeons from TYP in other campaigns, as well as LMoP. Therefore, if I run Dungeon of the Mad Mage, I need to search a bit afield – or just start them at level 5, which I usually don’t like to do. There’s something special in my mind about taking a character from a fresh-faced level 1 novice up to an experienced adventurer. The struggles in the first couple of levels make the middle levels feel even better.
Anyway, none of us should feel constrained to connect DH and DotMM at all. Other than both starting off in the same tavern, they don’t really have much in common. If you want to focus on dungeons, then do that! Alternately, if you want more capers and crazy antics, start with DH and then let whatever happens with your group direct where the action goes next.
Technoskald's Forge 