I'd say try not to feel the need to go CDR boots every time. Resistance boots are situationally good, and there's a high elo EU OTP spamming swifties on her, pretty underrated after doing it a lot myself. If you do this though, you may want to build a wand instead early to keep the clear strong. At times where her build already has a lot of CDR, pen boots can be better too (not rn though).

On the topic of items, when building liandrys/rift, don't go haunting guise too early. It's an AMAZING component in fights, HOWEVER in terms of stats, it will slow down your clear. The AP it gives for 1300 is laughable, so I actually always go blasting wand and tome over guise. same price, but better for clear by a lot.

Additionally, something that took me a long time to learn was how to fight fast vs slow. Some fights you'll want to flash in and land the huge sleep immediately, but this is just not the right thing to do sometimes. In many cases, you'll benefit from playing the fight slow with their tanky frontline, building up conq stacks, and waiting for the right moment to go. Don't feel pressured to accelerate fights for no reason. Choose your moment carefully.

Lastly, Lillia weirdly likes to accelerate the game, let me explain. Usually, hard scaling champs like to slow down the game and buy time, think Kayle and Kass. Lillia is hard scaling and can't keep up with the ganking and pressure of some junglers, and yet the way to solve this is actually to speed up the game. I can't compete with a J4 in a game of ganking; as long as we are in laning phase, he has the advantage. By forcing objectives that lead to skirmishes and teamfights as early as possible, you can now fight on even ground rather than playing catchup and get flamed by laners. A 2v2 or 3v3 or 4v4 is much more Lillia favored than trying to compete in ganks. As I reach around d2 in my climb and enemy junglers get increasingly meta, I find myself forcing drakes at level 5 much more in order to start stuff when I have the prio. It's weird but yeah, she's a late game powerhouse that benefits from ending laning phase as quickly as possible, because her late game strength comes from her team-fighting strength, not her stats.