* Fight against Islamist militancy also on Kerry agenda
* Tunisian uprising in 2011 inspired wave across Arab world
* Tunisia's advances contrast with setbacks in Egypt, Libya (Adds Kerry details throughout)
By Arshad Mohammed and Patrick Markey
TUNIS, Feb 18 (Reuters) - U.S. Secretary of State John Kerry on Tuesday praised democratic progress in Tunisia, where the "Arab Spring" began, and offered U.S. cooperation in the North African country's fight against Islamist militants.
During a brief visit to Tunis, Kerry said Tunisia and the United States would initiate a strategic dialogue, usually meaning regular high-level meetings, which would begin with a trip by Tunisia's premier to Washington.
After a crisis last year brought on by the killing of two opposition leaders, Tunisia adopted a new constitution and the ruling Islamists stepped aside for a caretaker administration to govern until elections.
Kerry's visit was to highlight progress since the 2011 uprising that brought down autocrat Zine el-Abidine Ben Ali as well as the compromises new Tunisian leaders, unlike Egyptian and Libyan counterparts, have made.
"We thought it was important to come to this country, which is struggling with this transition, in order to lend our support," Kerry said after meeting Tunisian President Moncef Marzouki and Prime Minister Mehdi Jomaa.
Tunisia's new constitution and steps to full democracy have been praised as a model in a region still widely unstable since popular revolts in 2011 that ousted long-standing rulers in Egypt, Yemen and Libya.
Egypt's military last year ousted its democratically elected Islamist president after mass protests against his rule. Libya is floundering in disorder with a weak central government unable to impose its authority on rival political factions and brigades of former rebels.
MORE ON 2012 EMBASSY ATTACK
Kerry addressed the many challenges still facing Tunisia, among them persistent violence by Islamist militants whose leader has pledged allegiance to al Qaeda's North African wing, officials said.
Kerry said he had discussed Tunisian requests for more assistance, following the U.S. provision of $400 million since Ben Ali's downfall, and that included more counter-terrorism aid though he declined to say what exactly the United States might provide.
"Most of the requests were focussed on equipment, although there is some thought about training and assistance in other ways, but that has to be discussed thoroughly," he said.
Tunisian security forces have been battling militants from the banned Islamist movement Ansar al-Sharia, one of the radical groups to emerge after Ben Ali's fall.
Ansar al-Sharia was blamed for inciting the storming of the U.S. embassy in Tunis on Sept. 14, 2012 and has since been listed by Washington as a terrorist organisation, with ties to al Qaeda in the Islamic Maghreb.
Kerry's visit to Tunisia is the first by a U.S. secretary of state since that incident, which occurred three days after an attack on a U.S. diplomatic mission in Benghazi in Libya killed the U.S. ambassador and three other Americans.
During meetings with the Tunisian president and prime minister on Tuesday, Kerry was also to urge the government to do more to arrest and prosecute those responsible for a 2012 attack on the U.S. embassy in Tunis, officials said.
While violence from Islamist militants is sporadic in Tunisia, which relies heavily on European tourism, militants have been using the chaos in neighbouring Libya to get access to arms and training across the porous border.
(Editing by Mark Heinrich and Robin Pomeroy)