I don't know what other people do, but I leave everything as-is while chilling. After chilling, since I'm a big strong dude, I have a friend hold a strainer as I dump the whole wort from the boil pot through the strainer into a bucket. Slam dunk! You've removed most of the large particles and you've managed to aerate a bit too. Then I dump it back and forth between two buckets three or four times to aerate. Then I siphon with a metal racking cane that has a filtering tip to ensure nothing larger than something the size of pencil shavings will be transferred to the fermenter. If you run your fermentation schedule properly your beer will finish up clear and pure as the wind-driven snow (unless you added wheat to your mash).
'Kilgore Trout once wrote a short story which was a dialogue between two pieces of yeast. They were discussing the possible purposes of life as they ate sugar and suffocated in their own excrement. Because of their limited intelligence, they never came close to guessing that they were making champagne.'
—Kurt Vonnegut, Breakfast of Champions