Eeeee *cringe*. Haha. Up to the lunch sounds like you did fine. Then:
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Just been to lunch with her. I looked for IOIs as recommended on this forum. She played with her hair, played with her ear asked about previous relationships. All good signs yeah?
Have you asked yourself why you are looking for IOI's? A lot of needy AFC's seem to do this way too much. I don't even consciously bother. Look at it this way. So you've noticed a bunch of IOI's. You're chuffed. Well, so fucking what? Noticing them makes NO DIFFERENCE to your at all to anything. It has no effect on your success. You actually have to act on it!! When did that part go out the window? It's as if some people just spend their lives vigorously hunting down IOI's with a microscope to make themselves feel better. Waste of time...anyway, that was more of a rant than it was aimed at you, rant over

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"if you'd like to do this again sometime then let me know" so the balls in her court. I think that was the right move?
Noooooo completely wrong move. What does that line mean, or what does it subcommunicate? This:
-I am too much of a pussy to be a man and invite you out next time
-I am needy because I need YOU to show me that you like me
-I have AFC written across my forehead.
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We went our separate ways with a friedly kiss
Haha, hate to break it to you bud but again that's typical (or worse still, stereotypical) AFC. A 'friendly kiss' = BS. Next time go for the lips, or a meaningful hug, or go in for a hug then reject her playfully then grab her, etc etc. Just do anything, ANYTHING that is unique. And don't settle for less than you deserve.
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Anyway. She's really great and I'd like to escalate things to a proper date and see where we go. But what I don't know is what to do next. Do I text her? Do I call her? When? What do I say/offer to do? Do I make it clear that I'm interested?
This is such a broad question, you're basically calling your entire game into question. My advice first is to study more, it's good that you're getting in-field practice, normally it's the other way around but you need some more theory before you chode yourself out of sets. Try and set up another outing which will go into the night so you can lay her.
Good luck buddy!