One Marine was killed and several others wounded on Thursday, when some 4,000 Marines launched the operation in Helmand province - a remote area that is at the center of the country's illegal opium cultivation, which helps finance the insurgency.
So far, however, there has been little resistance from the Taliban, according to a military spokesman Capt. Bill Pelletier.
In the east, meanwhile, U.S. troops and personnel continued looking for an American soldier believed captured by insurgents on Tuesday, said Navy Chief Petty Officer Brian Naranjo, a U.S. military spokesman.
U.S. troops were brought in from nearby areas to help with the search, which included helicopters and Afghan Army support, officials said Thursday.
Britain's Defense Ministry said a roadside bomb Wednesday in Helmand killed two soldiers, including the most senior U.K. officer to have died in combat in Afghanistan. Lt. Col. Rupert Thorneloe was the commanding officer of the 1st Battalion Welsh Guards.
British Prime Minister Gordon Brown praised Thorneloe as someone "I admire and someone who will be missed by all his colleagues in the armed forces for the leadership he gave, the bravery he showed and the professionalism at all times he exhibited."
The aim of the operation in Helmand is not simply to kill Taliban fighters but to win over the local population, Pelletier said - a difficult task in a region where foreigners are viewed with suspicion.
"We are not worried about the Taliban, we are not focused on them. We are focused on the people," Pelletier said. "It is important to engage with the key leaders, hear what they need most and what are their priorities."
The offensive along 55 miles (90 kilometers) of Taliban-controlled areas in southern Afghanistan will test the Obama administration's new strategy of holding territory to let the Afghan government sink roots in rural areas where Taliban influence is strong.
The insurgency has proven particularly resilient in the area, and foreign troops have never before operated in such large numbers there. Large areas have been under Taliban control, with little or no government presence.
As the operation entered its second day, the units secured control of the district centers of Nawa and Garmser, and negotiated entry into Khan Neshin, the capital of Rig district, Pelletier said.
"They waited for the local and village elders," outside Khan Neshin and "with their permission they went in and now are engaged in talks," Pelletier said.
As the Marines in the village of Nawa sat for a meeting with a group of 20 Afghan men and boys who were squatting on dirt ground, they listened as a list of their concerns came in a form of questions.
"Are you going to enter our houses?" asked 25-year old Mohammad Nabi, who was there with five of his younger brothers. "We are afraid that you will leave, and the Taliban will come back," he said. And they all described the police as predatory thieves not to be trusted.
Marine officers tried to reassure those around them they would not enter their homes and that they are here to stay throughout their deployment.
In a display of deep misunderstandings that any foreigner is at pains to overcome, an elder with a gray beard asked the Marines whether they will stop them saying prayers.
In describing the Taliban, they compared them to Americans.
"They spend one night in the village and then move onto another village, just as you guys," Nabi said.
Taking ground from the Taliban in Afghanistan has always proved easy. Keeping it and ensuring the government's presence has been the difficult part. The military challenges are compounded by the fact that the area is the world's largest producer of opium, and drug profits feed the insurgency and corrupt government officials.
Afghanistan accounts for more than 90 percent of the world's production of opium, and Helmand alone is responsible for about half that amount.
Haji Akhtar Mohammad, from Gereshk village now living in Helmand's capital of Lashkar Gah, said the U.S.-led force will not have community support in the region weary of any foreign interference.
"It is difficult to tell who is Taliban and who is civilians," Mohammad said. "They all have the same face, same beard and same turban," he said. "It is very difficult to defeat them."
Three years ago, only a handful of U.S. troops were in Helmand, Afghanistan's biggest province that is bisected by the Helmand river.
While Pelletier said winning hearts and minds was the mission's main focus, other military officials have said the immediate goal of the offensive is to clear away insurgents before Afghanistan's Aug. 20 presidential election.
Southern Afghanistan is a Taliban stronghold but also a region where Afghan President Hamid Karzai is seeking votes from fellow Pashtun tribesmen. Without such a large Marine assault, the Afghan government would likely not be able to set up voting booths where citizens could safely travel.
The Pentagon is deploying 21,000 additional troops to Afghanistan in time for the elections and expects the total number of U.S. forces there to reach 68,000 by year's end. That is double the number of troops in Afghanistan in 2008 but still half as many as are now in Iraq.
Even bigger challenges, perhaps, will come in the weeks and months after the Marines have established their presence here.
The U.S. will have an opportunity to help develop alternate livelihoods for farmers whose opium poppy crops bankroll the Taliban, who have made a violent comeback since the U.S.-led invasion ousted them from power in 2001.
In the country's east, meanwhile, a roadside bomb Friday killed three Afghans and a foreigner working on a road construction project, said Bismillah Mangal, the deputy governor Paktia province. The blast ripped through their vehicle as it was traveling on a road that connects Paktia and Khost province, Mangal said.
Also Friday, Russia announced that it will allow the U.S. to ship weapons across its territory to Afghanistan, providing Washington an alternative route to supply its forces in the landlocked country.
The normal route to Afghanistan via Pakistan has come under repeated Taliban attack, and the U.S. and NATO have been eager to have an alternate overland supply route through Russia and Central Asian countries.
Up until now, Russia has allowed the U.S. to ship non-lethal supplies across its territory for operations in Afghanistan, and Kremlin officials had suggested further cooperation was likely.
This program aired on July 3, 2009. The audio for this program is not available.