If you want to skip the long post below here is the TLDR version - Times (and laws) have changed, buying a car used to be a big hassle with a lot of haggle involved but it really isn't anymore. Deal with the internet sales department over the phone and walk in and sign the papers to close a deal within 1hr and almost no haggling now days.
LONG version - To get started I suggest you read
Confessions of a Car Salesman. This is an investigative journalism piece done by Edmunds.com back before they were well known. Some will say that this piece is what made their website take off.
What is generally conveyed in that piece is a bit dated, it was penned about 12y ago, and there have been a lot of changes in the industry since then (including regulatory reforms). But the basics still hold true. Most people are actually negotiating three separate deals when they buy a new car:
1. Deal #1 is the obvious deal. The price of the car you are buying. You should always deal in the purchase price of the car. Not the monthly payment. You are buying a car from the salesman, that's it. Usually this is where most people think the haggling and heartburn reside, but it isn't. Because of the proliferation of the internet you don't have to drive to different dealerships to 'price shop'. You can now pull up carmax, autotrader, cars.com, and competitor dealer websites right in front of the salesman and show competing prices. As such you'll generally get a pretty good price from any salesman who has more than two weeks on the job. It isn't worth their time to haggle on the price here because of all the easily available information out there. Besides, they can make more money on deal #2 that is coming.
2. Deal #2 is the hidden deal, and you start this before you drive up to your dealer to start #1 above. This is where the dealer and salesman actually make their money now days. Your trade in price. This price is highly subjective and fraught with a high amount of variability. There are thousands of dollars of commission hemmed up in this deal for the dealer. To get yourself a good starting point for negotiations you should prep for this though. Take your trade to the nearest CarMax and get a trade valuation by them. It is free and the best part is this is your ROCK BOTTOM number. Because CarMax will buy your car for this price regardless if you buy one of their cars. Once you have that go to two other mainstream dealers that sell the type of vehicle you are interested in. Looking for a Dodge 1ton 4x4, go to two Dodge dealers and, drive up to the used car lot and ask for a trade valuation to be done to your car. They'll ask if you want to buy something, tell them, "If you are going to give me a good deal on my trade, THEN we will talk about a purchase". But you won't, you are using them to get more negotiating firepower.
Now that you have your trade valued by three different places, you can talk with your dealer of choice about what your trade is worth to them. No need to be sly, just hand them your keys, let them know that you have already shopped your trade around and let them know what your rock bottom price is (CarMax). Use your other two trade valuations to back up the CarMax if needed, but you should be able to get above CarMax price for any trade at any dealer that deals in your type of used vehicle. Example, if you are trading in a 1ton 4x4 at a Dodge dealer, they should give you a better deal than CarMax because they deal in this type of vehicle and know how to move it. But if you are trading in a Porsche at a Kia dealer, CarMax will probably be a better deal, because the Kia dealer doesn't have the market to move a Porsche in (you can swap that around too, a Porsche dealer won't be able to move a Kia in the used car market). Just be realistic here and don't let yourself get pushed over.
3. Deal #3 is now pretty regulated and used to be another big hassle, but now days it doesn't have to be. Financing. Best bet is to get pre-approved through your bank/credit union and you don't have to worry about this at all. But let's assume you are going to go through financing at the dealer. In the past the finance manager used to have the power to manipulate the interest rate that you received on your loan. In the past the finance manager used to also get commission on that rate. This isn't the case anymore. This business sector has come under the ire of the government and is pretty well regulated now. The finance department is now mostly a broker. They just take your information to multiple lenders and show you your options. They generally get a flat rate for 'placing' that loan, and they make the same money no matter what the rate or who they place the loan with. BUT, that doesn't mean there isn't money to be made on the back end. They will talk financing with you, rates, whatever you want. But where THEY make their money is on the extras. Do you want that extended warranty, gap insurance, that clear bra, those running boards, etc. My standard line whenever I buy a car has always been, "I came to Dodge to buy a fucking dodge, not insurance and saran wrap. I don't want to see any extra line items on that contract. Fill the paperwork out and call me back in here when it is done." Then I'd go grab some coffee from the lobby and go back to their office.
Buying a car now days is nothing like it was in the 80's, 90's, or heck...nothing like it was just 7y ago. A lot has changed in the past 6y regarding car buying. It's not like buying a new pencil for your kid at school, but it certainly isn't like it used to be.