Dungeons and Dragons

You already had good answers but I would like to add my two cents:

The "starter set" is the cheaper option (less than 20$). It has a small prewritten adventure, a set a prewritten characters of level 1 with backstory, and the basic rules you need for this adventure.

Good point for it: cheap, the bare minimum you need to dive directly in it, already contain an adventure so it's easier for you as a new GM.

Bad point: it only contain a small subset of the rules, and will become "useless" if you decide to go further and buy the full books. Also, if your wife or kid do not like the pregen characters, you will not have the full rules related to character creation.

Then you have the core rule set already linked.

Good point: everything you need for a very long time, you will have all the rules to run anything you want. Other books are "only" going to add more options (spell, items, characters building options, ect).

Bad point: the price (120$), more information so it may be harder to digest everything and "get into it". It does not contain pre written adventure. You will have to find one separate or make one yourself, there are some free options available.

On top of that you don't need anything else exept pen and paper. Dice are of course greatly recommended, but you can start with some free phone app.

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!

There are two starter sets that are relatively cheap, the newest one is literally called 'starter set' and a slightly older one called the 'essentials kit'.

These don't contain full rules or player handbooks or anything, but are intended to provide enough to get going which is what it sounds like you need.

You can DEFINITELY play with just two player characters and a games master and I would strongly recommend jumping in to the latter role fully, without trying to also play along. It can be done but your players will be relying on you to keep the game moving and interesting, as well as playing all the other characters they will meet.

I've run the essentials kit and while it has its flaws it totally works as intended so I can vouch for that

I've started DMing with 3 friends a few months ago. None of us have played before, and we've been playing the Essentials Kit campaign. It comes with a simplified rulebook for the pre-made characters and some really good pre generated quests and story. I'd highly recommend it as a guide where you can either follow it completely or make up some side quests as you want.

I also picked up the Starter Kit but honestly it didn't contain enough for me in the beginning so we went with the Essentials Kit.

The rule box set with the Players Handbook, Dungeon Masters Guide, and Monster Manual is also good, however the only one I've used each session is the Players Handbook to give more context and allow the players to expand a bit, but it's definitely not required at least for the first few sessions.

This seems like the starter pack you're looking for: https://dnd.wizards.com/products/dd-core-rules-gift-set -- there is no Rules book, as such, just the phb and dms guide.

Magpie Games has a simplified table top game of avatar the last Airbender that I think would be more approachable for kids.

If you do decide to buy the full books the Player's Handbook should be your priority. I've been playing on and off with friends for awhile now and its all we've ever needed. The others are great for designing adventures but not really required for playing, and I've been able to find any statblocks I need online easily. But when its time to create a character, level up, or lookup a spell the handbook is great to have around.