Start with pre-made Characters
By which I don't mean generic characters that could be in any fantasy story, but by creating a character specifically for each of your children. These characters should be based around the characters from movies and TV shows they each seem to be drawn most toward. It's not the perfect fit that character creation is, but it can let you leap over the hurdles that character creation presents to a new player and let your children get right into the game. Your child's statement about it feeling exactly like a movie is what you want to most emulate.
Use essentials classes only, specifically MBA focused ones
Essentials classes (the Knight, the Slayer, the Warpriest for example) are built around having strong class features (always on) vs. making choices between how to use encounter and daily powers to most benefit the party. They work best in Heroic Tier (levels 1-10) and are strong classes that do not require as much optimization as the AEDU classes do. more importantly they focus on what are iconic archetypes both within and without of D&D. The Knight and the Slayer for example are both actually sub-classes of the traditional fighter. The first wears heavy armor and is all about protecting his friends while the second is about dealing as much damage as possible to monsters.
Use inherent bonuses
One of the best rules options you can take advantage of is inherent bonuses. Found in Dungeon Master's Guide 2, p. 138. as well as the Dark Sun Campaign Guide Book, p. 209 (where it was expanded) inherent bonuses take the place of magic items for the purposes of system math. At set levels the characters will gain +1, +2 etc. to their to hit rolls, damage rolls, and their defense stats. Magic items are still compatible in that their properties, item attacks, and bonuses to crit damage remain, but their mathematical bonus does not stack with inherent bonuses. Whichever bonus is largest is used.
Focus on the story and the adventure, adjudicate their actions to fit
If you were playing 4e with adults I would wholly say to depend upon the use of the powers their characters have and play the system as-is. However this may be a difficult pill for your children to swallow all at once (there are plenty of adults on the internet that can't handle 4e's separation of fluff, the descriptions and lore, from crunch, the hard rules themselves). When your children's turns in combat come up or they are making decisions out of combat ask them what they want their character to do and then based on their answer formulate what their character would do mechanically and then walk them through what their character does. This way you can introduce and have them take over parts of the rules at 1 piece at a time.
The Basic Rules are a place for you to start.
The Dungeon Master's D&D Basic Rules has a section on encounter building and modifying encounters (p.165-167) which will get you the same advice @Marius referenced in the DMG--at zero cost. Pay particular attention to "Party Size" on p. 166.
(The adventure contained in the Starter Set--The Lost Mine of Phandelver--does not contain notes on modifying it's encounters' levels.)
Note that the Basic Rules were revised in 2018. If you have the old Basic Rules, the referenced sections are on pages 56-58.
Next steps...
Take a look at Black Streams Solo Heroes, which has a lot of good advice for modifying adventures/campaigns/gm-style to suit the one-player experience.
Likewise, rpg.net has a good series of columns on Role-playing Duets that might give you some ideas.
But the best reference...
is to be up-front and honest with your players. Let them know you're not sure how things'll go and that you're very interested in their feedback. Some games are fun to play at the edge of survival, some players only want to play heroes who will embody the monomyth.
Most importantly:
All of the references above are distilled wisdom of other GMs over the years, wisdom borne of experience. It's all good, but none of it's going to be better than the experience you all get playing together and discussing what you like and what you hate. Use these as references for ideas, but use your players and yourself as your guide.
Best Answer
You need to read through the rule booklet, the Lost Mine of Phandelver adventure, and be ready to roll with whatever happens and have a good time.
They're standard ones for the adventure. Starting with them is probably a great place if you're not sure where to start.
If you want more options, you can download more "pregenerated characters" from the publisher's web site. (You can also download the "Starter Set Character Sheets" there, so you can print them out again in case one of the ones in your box loses a battle to a can of soda or you end up wanting to play the adventure again after you're done.)
If your players really want to customize everything, you can get the online Basic Rules and work through the character creation rules. I'd really advise against this approach, though, until your group is more familiar with the game. There's a lot of choices, and you need to play for a while to get a feel for things.
There are two main approaches to playing D&D (and options in-between): The first style is generally called "theater of the mind" style, where everything is just in your collective imaginations. The DM describes the action, and players imagine how their character would respond. You occasionally make judgement calls on whether somebody is close enough to be within range for an effect, but there generally isn't much of a focus on exactly who is where. The second style is more tactical, where you track where each monster and player is more precisely, generally using miniatures. Either approach can work great. If you're new, I'd suggest starting with theater of the mind and seeing how that works before going to buy maps and miniatures, unless you know that what's going to interest your group is playing with little miniatures. The adventure booklet has maps in it, but that just can be to help you visualize how big things are and what's where so that you can describe it to your players. Don't worry about exact locations of things unless you really want that more tactical approach.
If you really want to try out the maps and miniatures approach, find a grid to use (It can be something simple like printing out some sheets of 1-inch squares, or something like the back of some kinds of wrapping paper), draw rooms on it with pen or marker, and use random objects you already have as tokens (coins, Lego minifigures, pawns from other board games) to represent the player characters and creatures in your battles.
If you end up really liking the miniatures approach, you might want to get a dry-erase or wet-erase grid board. Wizards of the Coast publishes their own D&D Adventure Grid, but there are a variety of options from other publishers as well. (They don't need to be D&D branded, you're just looking for something with 1-inch-or-so squares that can be easily drawn on and erased.) As for miniatures themselves, there are similarly a lot of options from a lot of publishers, in a lot of price ranges. One option is just little cardboard squares that just show a position on the board. Another is full-fledged miniature figurines, generally made of plastic, pewter, or metal. (Some people enjoy a hobby of painting these as least as much as they like playing the game.) I recently bought the Arcknight Flat Plastic Miniatures DM Starter Set, which has printed graphics (including I think every creature you'll meet in the Lost Mine of Phandelver) on little plastic sheets which pop into little stands, giving a bit of three-dimensionality while not having the cost or taking-up-space-on-my-shelf that full miniatures would have. But all of that is "extras" I'd only recommend if you're looking for more things to get and if everybody's going to like using them; it's certainly not needed at all in order to play and have a great time.
Some people use digital maps, either because they're playing entirely online, or they use a large display or projector on or near their table and track monster and player positions on their screen. For Lost Mine of Phandelver specifically, the illustrator has digital maps of the adventure available for sale intended for this use, though one could print them out too if one wanted to mess with getting the scaling right and either put together multiple papers or find a large-format printer. Again, this is an option you don't need at all, but since you asked about what you might want to buy or use I wanted to include the possibility for completeness.
But even though I just spent a lot of words describing various ways of using maps, really, doing everything in your mind will work great. Only use something more complicated if your group will find it more fun and worth the cost.
The premise of the Starter Set is that it has everything you need to get started, and it really does. You may want more sets of dice, just so you don't need to keep passing the ones in the set around the table, but it's not really needed to start.
Sounds like you're off to a great start. Feel free to check out our chat room, browse questions here (see the new-gm and lost-mine-of-phandelver tags in particular), and look elsewhere on the Internet for tips, but really just don't worry about "screwing things up". Since sure you will make mistakes, and you'll discover things you wish you'd done differently, but the point of the game, like all games, is to have fun. Work on telling a fun story as a group, and don't let worrying, perfectionism, or even the rules get in the way of doing so.