Part of why it's so hard to DM Out of the Abyss is because it is a branching adventure- characters can go from Velkynvelve to just about anywhere, including Menzoberranzan, at 1st-level, right after the first chapter (talk about "out of the frying pan"...). I encourage the DM to read the first half of the book, at least, so he/she is acquainted with what is going on in each location that can be visited in the first half of the adventure- you never know what the players might think to go! That being said...
Allow the players to choose where they want to go
but if you present the options right, the PCs will follow how the chapters are presented. This is a hidden railroad trick that most players will not even notice, and I've personally used.
In the module itself, there are dis/incentives from going off the "ideal" path. If you haven't picked them up yourself, here are some that I have used:
- Travel Time - travelling from Velkynvelve straight to Gracklstugh, for example, will take 28 days of equipment-less, resource-less travel. Without food and water, that's simply too risky for low-level characters to attempt. If the PCs ignore this, make sure one of the underdark-savvy NPCs like the Drow or the Dwarf voices out the risks.
- Convenience - Having next to no equipment and resources, the PCs will want to re-supply, equip themselves and rest before going off again. Several locations are conveniently placed in between each other to allow the PCs to simply stop by, Sloobludop comes to mind, being a mere 8 days from Velkynvelve.
NPC motivation - this is one of the best incentives to get the PCs to go to locations. All of the NPCs that ally with you have strong motivations to go home to their respective cities, you can use this as a powerful tool. In my game, the PCs were foreshadowed early-on of the events in one of the locations
having met and killed the Parade of Fools in the Whorlstone Tunnels. By the end of that encounter, they wanted nothing to do with Neverlight Grove.
But when one of their closest friend-NPCs started voicing out their concerns about wanting to go home, they all agreed to go there anyway. This option, in particular, may sound like I'm forcing them along the rails. To an extent, that is true. But I talked with my players about the option, and it turns out that they really wanted to go because it was interesting and exciting, despite the apparent danger. I pushed them, sure, but they were motivated enough to go there already, they only needed a little push.
If, despite all of the above, they still stray from the "ideal" path, then it's a good thing you've read up on the location before hand, as I encouraged. If you haven't, you still have the time to catch up on reading while you have a travel session where the PCs face the perils of the underdark.