So first I do my homework. Get my own financing. Know exactly what model I want. Then I look online for all the dealers selling this vehicle, note the place, contact info, leave room for the name of the rep I'll be speaking to, price, etc.
Then I call each one, telling them I saw the online inventory had the vehicle in stock but since I'm buying today I just want to check if that's still the case. That already reduces some of the options: many dealers will leave attractive but outdated inventory to generate leads, and then try to upsell.
I start from the cheapest option I see online and work my way through the list. As I encounter the higher priced versions, I voice a slight disappointement that another dealer X is selling the same model for $Y. That's when they can choose to match or stay firm. Either way I thank them for their time and tell them I will be calling back within a couple hours to let them know.
During this time, I often get callbacks with counter-offers. I note the value, thank them and let them know I'm finishing my list.
Once I have a winner, I go back and call all the others, starting with the most pricy, to let them know I've made my decision, and that I respect their time so didn't want to let them hang. Things get interesting at that stage, as more counter-offers appear. At that stage, you can also visit in person to thank the salesperson as you're on the way to the lowest offer: sometimes they end up with yet another counter offer.
Having your own financing means you're not stuck with their shell game. Only when you go to doing the paperwork, you can negotiate financing rates and see if they can beat what you got from your bank. But the sale price is already set at that stage so it's safe.
The whole point during this process, it's super important to remain humble and friendly. Even though you control the process, any smugness will kill your options. And to tie this back to the thread, that's a skill you can improve with negotiation training (and it benefits more than just car buying).