"Regarding a truce, a ceasefire, the delivery of aid, this is a moral question and there is no debate around this, we absolutely welcome this, but there are other articles around which there are reservations," Zakaria Malahifji of an Aleppo-based rebel faction told Reuters.
Combatant sources on both sides said calm prevailed in the first hours of the ceasefire but reported violations increased later in the night.
Russia is a major backer of Assad, while the United States supports some of the rebel groups fighting to topple him.
U.S. Secretary of State John Kerry said early reports suggested there had been some reduction in violence.
He told reporters at the State Department it was too early to draw a definitive conclusion about how effective the truce will be and that there would no doubt be some reports of violations "here and there".
The agreement's initial aims include allowing humanitarian access and joint U.S.-Russian targeting of jihadist groups, which are not covered by the agreement.
Russia's Foreign Ministry said in a statement that aid to the besieged city of Aleppo would start immediately.
Besher Hawi, a resident of the opposition-held city, said it had been calm since the ceasefire came into force, after a heavy day of bombardment.
"It's excellent but I certainly have no confidence in the regime. It could bomb at any moment," he told Reuters from Aleppo, speaking via a web-based messaging system.
Residents of government-held western Aleppo, frequently hit by rebel shelling, also expressed doubt over whether the truce would last. "Every time there's a truce the militants ... hit us," said an Aleppo resident who gave his name as Khaled. "We hope things will improve. May the army be victorious."
As the night went on, the Syrian Observatory for Human Rights, a Britain-based monitoring group, reported government air strikes and bombardment in Hama and Aleppo provinces, and shelling of rebel-held areas near Damascus, though it said the violations were not serious. A rebel in the Aleppo area said warplanes had opened fire with machine guns north of the city.
A Syrian military source meanwhile said armed groups in Aleppo had sniped on residential buildings, and fired three mortar bombs at a government-held area on the city outskirts.
The ceasefire comes at a time when Assad's position on the battlefield is stronger than it has been since the earliest months of the war, thanks to Russian and Iranian military support. Hundreds of thousands of Syrians have been killed in the conflict and 11 million made homeless in the world's worst refugee crisis.