Regardless of what others do, stick to conventional driving rules and keep cool. Keep within the 100 km driving speed, use your turn signal all the time. You may have to pass on the right, this may be unavoidable.
Consider using Waze. The Spanish language version has more details. If you are traveling with another person, use both -- one in Spanish, one in English. It is best to take the Santiago Circunvalaci?n. Unless you want to stop in Santiago to visit the Centro Le?n, the Santiago cultural center that gives a good review of Dominican culture in general. You could snack at their cafeteria.
Another option is to head for Bonao and have a meal at the Tipico Bonao (there are two -- you want the one to the south -- it shows as Av. Libertad), you have to take a right turn, but it is indicated. They have good Dominican food. They are usually full but service is fast. If they are full, register at the cashier and they will shortly get you a table.
Waze will guide you to pass through Santo Domingo. There are many routes, but you need to go East.
You also could make a stopover in Santo Domingo, to tour the Colonial City. It is on the way. You could have lunch at Buche de Perico. Then it is back on the Las Americas Expressway all the way to the East. If you want to stop midway, there is Meson Espa?ol in Juan Dolio for delicious seafood fideua.
You also could make a stopover in La Romana -- but if you want to continue, make sure to keep in direction of Higuey and not Casa de Campo. If you have indicated Punta Cana on Waze they will guide you to another express toll road and keep you from entering La Romana town.
If you need a break on the highway, there is Shish Kabab in a gas station right on the highway. It is a good place to stop. Then it is all the way to Punta Cana.
Take note that while there is lots going on from Puerto Plata to Santo Domingo -- lots of attractive goods to buy on the highway -- there is practically nothing from La Romana to Punta Cana -- even the scenery is boring. That is why it is good to make the stop over in Shish Kabab (they are famous for their "quipes" and Arab food Dominican-style) to keep your driving relaxed.
Take care!