One Nato soldier and four Afghan civilians were also wounded in the first attack, which took place in the centre of Kandahar, a Nato spokesperson said on Monday.
Nato soldiers cordoned off the site of the attack, where a destroyed Nato vehicle and rising smoke could be seen, witnesses said.
Another suicide bomber blew himself up in a car near Kabul's international airport, injuring three people, eyewitnesses said.
Pieces of the vehicle and human flesh were seen strewn across the blast site as police and Nato-led troops sealed off the area.
Hayat Khan, who describes himself as a Taliban commander, told Reuters by telephone that both attacks were carried out by Taliban bombers.
In Ghazni province, alliance forces surrounded a suspected militant hideout, drawing fire from inside, the US military said in a statement on Monday.
One soldier was wounded before the forces called in air support, it said.
The troops found the bodies of three suspected fighters, plus bomb-making materials inside the compound, the military said.
Although Nato has assumed control of military operations in Afghanistan, the US is leading a smaller coalition of a few thousand troops to focus on "anti-terror" operations and the hunt for Osama bin Laden.