Hello there and welcome, congrats on adopting your bunny

When introducing leafy greens, less is best. A rabbit that isn’t used to eating them doesn’t have the correct stomach bacteria to be able to digest them properly which can lead to mushy poops, and painful gas. I would start with either the romaine or the cilantro, as both as commonly well accepted by rabbit stomachs. Don’t start with more than one type at once because if there’s any upset, you won’t know which green has caused it. In terms of quantities, I would give a piece of cilantro or romaine that’s about the size of a finger. The general rule is to then wait 24 hours to check for stomach upset, and if there’s no problems, double the size of the piece you give the next day. After about 5 days with no problems, you can introduce the next green.
This is certainly an overcautious method, and some people get away with feeding more than this first off, however my Bandit was quite sensitive to new greens and anything more than a small piece for the first few days would upset his stomach.

In regards to carrots, the root themselves should be regarded as a treat only as it’s high in sugar, so no more than a 1 inch piece daily, and I would hold off a few days until the first introductions of greens has happened. In addition, carrot greens are high in calcium, high calcium greens should be alternated out every other day, the same with greens that are high in oxalates. Binky Bunny has a great list detailing high oxalic greens and recommended feeding for treat type veggies: https://binkybunny.com/BUNNYINFO/VeggieList/tabid/144/Default.aspx