My daughter is 10 and interested in playing so I bought her some dice for Christmas.  I am a very experienced DM (over 30 years experience) but have never run a game for someone so young.  I'll be easing her into the rules, helping her create her character step by step...  I know enough to try and run the game that she wants to play - ie I'm not going to force her to go take care of a cavern full of kobolds if she just wants to fly around on a pegasus... but I was wondering what other helpful tips or pitfalls other parent/DMs might be willing to share.  I'm excited to finally share the hobby with my kid!

In all honesty  the best thing I can say is let her play  her character  when I was 4 my dad let me play Puss in Boots  and my sister play  Cardcaptor  Sakura  at his table. It was really  fun for both of us .  It's fine to be a little  goofy . Don't try to  be  a web dm  just  have fun. That is  my advice. 

I'd say... don't try to teach them the rules ahead of time. Just let them tell you what they want to do then help show them what they need to roll to figure it out. If they say something ridiculous, just try to find whatever skill or ability their character has that's as close as possible to what they're trying to do and have them do that instead. It also depends on the kid... some 10 year olds will pick up the game right away, especially if they watch D&D streams or if they've watched you and your friends play in the past.

Be patient. Be flexible. Let your daughter help shape the world and campaign and don't overload her with every rule at once. It's okay not to be 100% in accordance with the rules if you and your group are enjoying the game.

What is your daughters favorite movie, book, or fictional character?

Create an adventure around that.  

It does not have to be hack and slash RPG, but fun and light hearted.  Like she can bonk goblins over the head with a magic wand and the disappear and a cloud of glitter. 

Running a game for children is always a lot of fun.  There's a bunch of good advice here!

As has been said, a lot will depend on the individual child.  They might surprise you with what they want from the game.  Just try to be very flexible and check in with them a lot.  It doesn't have to be really obvious.  You can just say things like 'oh wow, that monster looks pretty scary, huh?'  They'll let you know if the whole scene is too intense.  Or they might just laugh and charge into to fight it.  Or they might say it looks sad and they want to help it.

Kids will give you so much to run with in a game.  You can just show them a map or a picture and they will start telling you what they think is going on.  I find they usually want the reality to be very close to what they expected, but with a twist.   (This is true for running games for adults too haha)

Things that most kids like -

A map (either of the local area with interesting spots highlighted, or of the specific locations they are in, like a battle map)

Props - if you have miniatures they love to play with them. They don't even have to be close to what they represent.  I've used so many toys to represent monsters and NPCs.

Pets - I have a rule for pets.  They're totally invincible.  For young kids, the pet never gets hurt and the kid gets to say what they do and control them if they want to.  If they don't want to control them, then I'll play the pet like an NPC that is completely friendly and loyal.   For older kids and adults, I give them an option.   If the pet is going to be used in combat, or for a frequent game benefit like a familiar, then it can be hurt.  But if you just want the pet for a friend, then I'll never target it.  It will survive anything your character survives.   It basically has plot armor.

Luck - some kids hate to roll badly.  But they also don't usually like to do anything they consider cheating.  So if they know they rolled bad, they don't want you to just tell them it doesn't matter.   So I like to give them something that feels like a part of the world.   A lucky necklace, or a wishing stone, or just a pool of inspriration tokens.  Tell them they can use it for a re-roll.  It could be limited to once a day, until the tokens run out, or even once a turn. Whatever feels right to you and them.  Just don't change the rules on how luck works mid game.  They'll think you're just letting them win.  If their luck runs out and they are taking it badly, you can always still let them 'fail forwards.'  Have the failure have a bad effect, but also help them into a better place next turn.  Like missing an attack, but lunging so far they get behind the monster and it doesn't know where they went. 

I hope you and your daughter have a really great game!

I'd recommend Matt Coleville's Delian Tomb as a first adventure.  It is a very simple four room dungeon and I think you fight four to six goblins and that's it.  You might play it or something like that with her one on one, and then see if she has any friends she wants to share it with.

You might be in for a bunch of pink unicorns and stuff for a while.

Is she old enough to read The Hobbit?  Would you want to read it to her?

Make it an interactive experience between you and her. Let her shape her character, any companions she has and to some extent the world around her so that she feels happy with them. I started in middle school so by that time I was mostly out of the kid stuff but I can still empathize.

I actually had her create her character's look on HeroForge (not going to commit to buying the mini quite yet) and it gave me a lot of great fodder.  She chose a dagger with 3 feathers on the pommel and painted it to look like crystal.  She chose a cat as a pet.  I'm basically going to take all of the choices she made for her mini and use them to create the stories for her.  I think each feather is going to have to come from a different magical creature: pegasus, griffon, maybe a couatl....  the cat might come from the Feywild....  

Those all sound like great ideas. Encourage her to create a personality for her pet too. Pets are amazing companions. My first character’s pet was a fey cat (a cait sidhe) too.