I'm sure someone more experienced than me can give you more confident suggestions but I've been seeing the Starter Kit, Essentials Kit, and Core Rules Gift Set advertised more for the holidays. The descriptions make some of them sound like massive expansions on the bare basics and thereby possibly more overwhelming than you'd probably need at first. I think some include location books that may be different from the kinds of environments you want in your own campaign too so be sure to examine what's included.

I've mostly only played and DM'd in homebrew groups so I'm probably too loose with it all but I will say you definitely need those two things you mentioned. Here's what I found necessary to really get things started:

• Players Handbook
• DM Handbook
• Monster Manual
• DM Screen (don't overlook this one, there are a bunch you can purchase that not only keep your players from reading off your game script but also have quick sheet information that is incredibly helpful on the fly. You can also print your own for free if you can create or find what you want, the DnD Compendium here has some free DM Screen images and files that you can print.
• Large Graph Paper (to draw your environments, and to measure how far players/NPCs can move per turn, etc.).
• Mini Figures (you can custom make some to fit your family's characters at something like Hero Forge, buy different pre-made characters and monsters, or go the cheap route and print out pictures of characters that you can fold so they stand up on the board).
• Dice. Or, alternatively, I know some DM's that run their own dice program code on their laptops so they don't have to physically roll for monster and NPC actions and can just run it through their programs (but if you go this route, still get your family their own dice sets so they can roll for their own characters' actions).
• A laptop with your game script and a darn good soundtrack playlist ready so you can set the right music mood for battles, taverns, etc.

You can also pick up a campaign book that interests you, or find some of the cool free one-shot campaigns right here in this very Lemmy community to cut down on the world and quest building! And of course, before I waste any more of your time, make sure to check out the DM Academy Lemmy community that's all about teaching and learning how to DM!