I noticed somewhere on the boards that there's a cap on cooldown reduction, I think it was 60%. Are there similar caps for, say, attack speed or ability power, or any other stats?

CDR is capped at 40%, Attack speed is capped at 2.5, movespeed has a soft cap of, I think, 430 and a stronger soft cap at, I think, 490. Spell/Armor Pen has an effective cap of whatever your opponent's Armor/MR is. Nothing else I can think of.

Thank you!

And while we're at it, I have another noob question: 2.5 for attack speed means 2.5 hits per second, i think? and movement is units per second?

not sure about movement but yes 2.5 is 2.5hps

anything with unique passive doesn't stack.

Example, two thornmails = 200 armor, but only one of the passives will work.

Tenacity doesn't stack either. If you have multiple items that give Tenacity only the highest value will have an effect.

To add to something similar than Tenacity, 'Enhanced Movement Speed' from the various boots will only count the highest amount.

movement speed only has a perceived cap, much like magic pen. you can get as much movement speed as you want, but most if it ends up being from buffs and such. magic pen has two perceived caps, you can only get about 50 magic pen from items and runes, another 20 reduction from abyssal scepter, and 49 percent magic penetration, unless you are xerath, in which case i believe you can hit 66~ percent roughly. theres a lot of false caps that are only there because of items, and many stats stack multiplicatively so its not great to stack them too high, but any other questions you have, the new player forum can help you out

For movement speed:
Any buffs beyond the highest buff are reduced by a large amount.
There are two thresholds beyond which additional MS is reduced.