> If I use an IP:Port proxy server to connect to the internet, I'm aware that the destination server (website) doesn't exactly know it's me, but can my ISP know what site I went to?
Yes, of course. And if your proxy server pattern is in any way predictable, and in particular if you ever visit a site where you need to sign in, then they have everything -- who you are, and where you went.
Also, a proxy server doesn't undermine cookie tracking. The cookie scheme doesn't care what your IP is, only that you have the right cookies, so this undermines the goal of anonymity.
Also, if for the sake of argument you encrypt everything, and if later on the police want to know what actually happened, they can try to force you to unencrypt your communications (or grant permission for experts to do so). In a recent landmark case, the police tried to make a child pornography suspect provide a password to unencrypt what might have been incriminating files of alleged child pornography. This effort ultimately failed (and the police decrypted the drives by brute force), but it indicates the direction things are going.
http://www.wired.com/threatlevel/2013/08/feds-crack-encrypte...
> Lastly, if my ISP can actually see what I'm doing, is there a way to hide from them (e.g. does connecting through an HTTPS proxy hide what I do)?
There are intermediaries that scramble your transactions, encrypt them, so the content cannot be monitored. Others randomly route your traffic in such a way that it becomes very difficult to trace. Here's one example:
https://www.torproject.org/
As to https, it's a suitable level of encryption for everyday matters, but it's not unbreakable, so if there's anything that might come back to bite you, don't rely on it.