Led by a Korean War-vintage tank manned by a guitar-strumming gunner, Tanzanian infantrymen today launched a long-expected offensive to open Uganda's economic lifeline to Kenya and impose government authority throughout the country.

Exactly one week after securing the capital of Kampala, about 28000 troops marched without opposition eastward toward Jinja in what their officers described as a month-long campaign to reestablish law and order in half the country now without constituted authority.

Reports from Ugandan refugees crossing into Kenya to seek asylum spoke of stragglers from ousted President Idi Amin's atomized army committing wholesale massacres in Tororo and other areas of the country. Andrew Ouma, 21, told newsman here that he had come from near Jinja this morning and said he heard reports that Amin's troops were looting the city and beating up shopkeepers.

Maj. Cyril Okido, commanding the 7th Battalion leading the Tanzanians slow-motion drive, said he expects to reach Jinja, on the eastern banks of the Nile River where it flows into Lake Victoria, by early Friday morning.

"We are not going in a hurry," said the jaunty major as he marched at the head of his troops through this bus-stop crossing, six miles beyond the previous easternmost Tanzanian position near Kampala, but still a good 30 miles from Jinja.

Other Tanzanian officers said a second column would be dispatched due north from Kampala and eventually link up with the Jinja-bound troops.

Okido's column was expected to march to the Kenyan border, thereby allowing renewed truck traffic to bring in petroleum products and other essential goods for that half of Uganda under Tanzanian and provisional government control.

Once those additional 75 miles due east from Jinja are cleared, the column will swing north to Tororo, Mbale and Soroti before heading west to meet the other Tanzanian troops at Lira in northern Uganda, the officers said.

Although few of Amin's stragglers were thought likely to attack the Tanzanians, the slow pace, which has distinguished Tanzanian tactics since the invasion began in February, was dicatated by a lack of transport and the thick undergrowth and forest which made the line of march ideal ambush country.

Aside from an occasional requisitioned truck, a half-dozen Land Rovers, a handful of armored personnel carriers, and three Soviet-built tanks, two of them dating from the World War II, the Tanzanians moved on foot.

They carried an assortment of weapons-Soviet-designed AK47 submachine guns, B40 self-propelled rockets, recoilless rifles and a half dozen towed antiaircraft guns-as they streamed along both sides of the main highway, their faces glistening with sweat in the hot sun.

Little boys offered them water and sugar cane and Ugandan civilians cheered as they trudged along.

Along the line of march were troops carrying umbrellas to ward off the sun, others sporting "liberated" wrist watches, still others wearing army great costs despite the heat and many carrying ammunition boxes, transistor radios and tape decks on their heads.

In Kampala, Dr. Martin Aliker, adviser to provisional President Yusufu Lue, told reporters Amin was last seen Sunday in Soroti, about 240 miles northeast of the capital, when he left by plane for an undisclosed destination generally thought to be Libya.

Aliker said he learned of the departure from two of Amin's Ugandan pilots who turned themselves in, one by flying to Kilimanjaro Airport in Tanzania, the other landing with his family at Uganda's Entebbe International Airport.

At least 800 Ugandans who arrived in Kenya since the fall of Kampala have been taken in for questioning, including military and government officials and John Akii-Bua, a former Olypmic hurdling champion, officials said.

Other official sources confirmed that Robert Astles, Amin's top aide during his bloody rule, was being detained in Kenya for interrogation. At one point Astles was reported to have been killed during fighting in Kampala. CAPTION: Picture, Soldiers of the new Ugandan government watch as fire destroys a warehouse in Kampala after looting there. AP; Map, no caption, By Richard Furno-The Washington Post