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I did something really weird on my last DM run of GDQ.
I ran D1 and D2 first, as a follow-up to the Slavers (A1-4) series. Why? Because fundamentally, D1 and D2 are a whole lot easier than G1-3.
Think about it: a bunch of trogolodytes, some bugbears, a warren of trolls, and a horde of kuo-toa. Most encounters in D1 and D2 can be readily handled by area-effect magic. There is exactly one encounter where it's time to break out the Brown Trousers: a certain fellow who's napping in an illusionary niche and simply doesn't want to be disturbed. And taking him on is entirely optional.
Compare with G1-3. G1 can go off the rails quickly. Walk through the wrong door, and you're facing a couple thousand HP worth of hill giants. Multiple encounters with 4-6 frost or fire giants. Remorhaz, pyrohydra, manticores, packs of hell hounds - basically, massive energy damage madness. A pair of white dragons, which is bad... and Brazzemal, an ancient red, which is even worse. The single toughest drow in the series behind a deadly magic barrier. Two lost temples (G1 and G3) where bad decisions can result in permanent character loss. Mind flayers. Named NPCs with built-in rules that can one-strike PCs (chuck 'em in the iron maiden, toss 'em in the lava, decapitate the kneeling fools).
So, run D1-2 as a stand-alone. It's an initial foray into the underdark. Downplay the drow elements. The end-goal is something related to D2 (loot the treasury! visit Blibdoolpoop! slay Va-Gulgh! free a prisoner!). Once that's done, it's time to crack out the classic G1-3, and may the gods have mercy on the party.
If you want to do D3 (the weakest of the bunch, in my opinion), the tunnel at the end of G3 leads directly to the Vault. Alternatively, replace the tunnel with a portal to the Demonweb Pits and skip straight to Q1.
That worked out pretty well for me, using either 4e or 5e. The masses of lower-level opponents from D1-2 tied in nicely to mid-level PCs, then the party is challenged by tougher giants and dragons and whatever, and then you end up with demons - which tend to be pretty nasty in any rule-set.