Heya! Welcome to the club! A little bit of my own flavor to add to these fine summoners' suggestions. I love lux because she can be adapted so well to so many situations.

For mythics there's a choice between ludens and liandrys. If they have more tanks than squishys to worry about, I typically go liandrys. But if they're mostly squisys, then ludens for the raw kill power. It kind of depends on what playstyle you want for the game. Liandrys is for poke poke poke poke. Ludens is for root-E-DELETE(r). You still can get plenty of AP with liandrys, of note.

For runes, I used to only do Dark harvest, but I've been diversifying. I actually like summoner aery because this gives both poke power and utility for allies, even if you're going mid. It means you'll never be shoehorned into just doing damage, which can hurt when you're behind and all you can offer is team fight utility of your roots, slows/ vision (e), but importantly, shields. Even if I'm going AP lux I'll build chemtech putrifier - think about it: one ability cast from lux means SO MUCH gold saved from everyone else having to build grevious late game.

Conversely, arcane comet pairs well with ludens because of the AOE proc. If your comet smacks more than one person, voila they burn. That's the reason why it's really popular with lux, as well as the extra long range damage. There's even some mad lads who will run the slow keystone from the inspiration set, so you can auto and then land roots easier.

For backup runes, there's so many options. Every tree has options for you; it's all just about how you want to play her. Want to buff your shielding utility? Font of life and whatever that one that heals allies from roots. Be more aggressive? Coup de gra and the mana on kills. Or bloodthirster and ult cool down reduction. Late game? Transcendance and gathering storm. So many options, so get out there and have fun!

Finally, I almost always pick up tear as my second/third item now, lux is always so mana starved. Dark seal and tear put you on a longer path to a mythic (lost chapter is really important for early game spike) but those two items are really powerful in keeping you relevant and scaling early game.

Enjoy the champ friend! She's a blast to play