How to quickly check your mining speed? Is there some website that provides some webminer or other way for a person to check their mining speed without actually joining any pool or doing solo mining? Alternatively, are there some miners that can be pre-configured to perform mining on a specific pool with given credentials?

Or to put it in another way, if I wanted to simplify the process of checking a mining speed of a given computer for someone with minimal knowledge of what Bitcoin is, what websites or software should I use?
Very good question. Today I created a page at my pool for this purpose: http://bitminter.com/test

You don't get paid without registering and entering a name and password. But it is useful for the purposes you stated in the question. Just click the "test start" button. You need Java installed, and of course GPU drivers with OpenCL support if that's what you want to test.

PS: Thanks for the idea for this.