Your right in that you connect to your ISP first, and proxy second. Heres a basic version if how the connection looks:
You ---> ISP ----> proxy server ---> desired webpage
But since your ISP/school/whatever only blacklists the desired webserver, the connection to the proxy is allowed as all they see is you ---> proxy and back. Then the infro from proxy ---> blocked place is hidden from them.
Then when served from blocked page ---> proxy, it encapsulates that data within packets from the proxy ----> you. At least, im assuming its encapsulated...cant think of how else it would be delivered...
So no, proxies arent totally 100% perfectly anonymous as anyone interested can do a whois on the proxy your connecting too, then try to subpoena logs from them to convict you or w/e. But thats where the art of choosing your proxy comes in, chain together 2 or 3 from 3 seperate countries, and bonus points if their not friendly with yours, then you have a damn good chance of not getting caught. Because for LE to get authorization to look at logs in another country will take months to a year, and much much longer if theres more than one country involved.
Hope this helps man, sorry i couldnt be more exact with the technical details :/