Once you're ready to negotiate in earnest with a salesperson, you'll probably be brought to a "closing room" off of the sales floor. If there's a choice, though, ask to stay in one of the cubicles in the showroom. Staying out in the open is generally less intimidating when you face an intense salesperson.
Start by making the first offer, which should be at or a little above the vehicle's invoice price. Tell the salesperson how you arrived at this price. Explain that if he or she can meet it or come close, you'll close the deal on the spot. Typically, the salesperson will respond with a counteroffer that will be slightly less than the vehicle's retail price. Next, raise your initial offer by incremental amounts of, say, one or two hundred dollars at a time. With each new figure, the salesperson will likely lower his or her price in the same manner.
At some point, the salesperson may leave to "present your offer to the sales manager" (though he may not actually do so). He or she will probably return with another bid. If it's close to your last offer, try standing firm; but if it's considerably higher, continue the negotiating process.
If the salesperson goes to "see the manager" a second time, his or her next counteroffer will probably be the dealership's lowest price on that day for that model. At this point, decide whether to accept the offer, leave and try another dealer, or keep negotiating.
Once you've reached an agreeable offer, bring up the subject of your trade-in and negotiate that price separately. Getting the absolute lowest figure isn't always worth the aggravation, and can lead to hostility that may be troublesome later. If you show a little willingness to accept a close offer, you may be treated more courteously when you have to return to the dealership. That's human nature. Paying a few dollars more than rock-bottom isn't a tragedy when you're talking about a purchase that runs to many thousands of dollars.