new-gm
I believe that all tabletop role playing gamers should at one point or another try their hand at Dungeon Mastering. However, it's a goal that at times can feel overwhelming.
What advice would you give to a gamer who feels motivated enough to want to run their own games, but are unsure where to start?
There are a variety of resources nowadays that can help you accomplish this. There are also many existing questions on this site about GMing that will point you to more content than you can ever consume.
In your question, you mention wanting to see more examples of real play. There's a number of ways to do so.
Actual Play Resources
Some games also have better advice sections than others - see What role-playing games have good gamemaster advice sections?
In the end though just watching is not the most effective approach to learning. Watching games is less useful experience than actually being in one. Have you considered playing in those games before running them to learn from other GMs? It's reasonably easy to find other gaming groups, you don't have to abandon yours to play in another. Where can I find other RPG players?
Go to RPG conventions, find games at gaming stores, play on forums or G+ (see also Finding online RPG players for a play-by-chat RPG Campaign?) - just get more experience. The GM was often called the "judge" in the old days, and in the legal world you need to spend a lot of time being a lawyer before you make a good judge. You need to spend some time playing to become a good GM. If you can't think how the players will proceed in a given situation, you need more play time.
There are many books on GMing - see What is the single most influential book every GM should read?
Also try watching/reading relevant genre media. "I don't get how to put together a story" should get its first-order correction by consuming some of that genre and looking at the stories.
A lot of the problem you seem to be experiencing is pure storytelling. Try How do I get better at narrating/storytelling as a GM? and As a GM, how can I create and role-play diverse NPCs better? Read up on the specific aspects of GMing you feel you're not good at, there's plenty here. Try questions tagged with the gm-techniques tag. Feel free and ask questions here as well about specific aspects of GMing.
There are also a large, large number of RPG forums out there in the world, for every game and type of gaming. If you don't understand something someone posts, you can easily reply and ask.
aka How I Learned To GM
We didn't have these newfangled Interwebs when I was a kid. I GMed almost before I ever played. I did play in a very informal game of D&D in a car on the way to Scout camp, no dice, PvP, everyone had artifact weapons. But other than that, I started out as a GM. I bought a sci-fi RPG (Star Frontiers) without knowing anything about it (I had bought and played a little TSR chit game, Star Force, and was looking for other fun stuff from the same company). None of my friends were interested in GMing and I was in a small Texas town that didn't have conventions or whatnot - life was less mobile and connected back then. So I just read the game books and then ran games for my friends. And I kept running them, and learned from my mistakes and corrected. I read comics and science fiction avidly, so characters and plots weren't that hard to devise. Beyond that, I just learned the way you learn to do anything through practice, whether it's a sport, writing, a musical instrument... How-to's and YouTube videos are cute jumpstarters nowadays, but "Do, and learn from doing" has yet to be eclipsed in being the primary way to actually become good at something.
Fear of "making a mistake" is the dumbest and most paralyzing instinct you can have in life. In a video game you're going to die a couple times off the bat; in baseball you're going to swing and miss a lot before you hit; in baking you're gonna burn some cookies. But you learn through those mistakes. It's fine to do a little reading up ahead of time but the only way to become good, really, is get your butt in gear and do it.
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.
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.
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.
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.
Best Answer
I would keep in mind the following:
Don't panic!
You are almost certainly doing a better job than you think you are
Read the rules
Start small, a single adventure
Consider using pre published material
Remember your job is to help the players have fun. Sometimes that might mean you have less fun, unless you get your personal kicks from happy players. Which you should do!
You do not have a character! Don't overshadow the players with your NPCs. Ever.