If you want to quote part of a website which includes graphics (so you can't just highlight it), here's a cool way of doing it! Go to the kwout link and enter the address of the webpage you wish to quote in the box where it says "http:// ...", drag a box over the region of the webpage you wish to quote, click "Cut Out", then choose the options that suit you and copy the code given to embed the image of the chosen area somewhere else on the web.
As for printing parts of a webpages, a very simple trick I only learnt recently is that of highlighting the part you wish to print, then when you print, click on "print selection only". This is a good way of avoiding banners, advertising etc. that usually plague the tops of webpages. Not all browsers have this option though, but some do, such as Firefox.
And one final note, suppose you want to save the selection, but not necessarily print it. Then, again you highlight and "print selection only", but you also check the option "print to file", so instead of printing, the selected part of the webpage will be saved as a file on your computer.