Mr Trump also warned Pakistan that the US would no longer tolerate the country offering "safe havens" to extremists, saying the country had "much to lose" if it did not side with the Americans.
"We have been paying Pakistan billions and billions of dollars - at the same time they are housing the very terrorists that we are fighting," he said.
He also said the US would seek a stronger partnership with India.
Meanwhile, Mr Trump made it clear he expects his existing allies to support him in his new strategy, telling them he wanted them to raise their countries' contributions "in line with our own".
US Defence Secretary Jim Mattis indicated in a statement "several" US allies had already "committed to increasing their troop numbers".
But Mr Trump refused to be drawn on how many extra US troops, if any, would be deployed. He had been expected to say another 4,000 would be sent to Afghanistan, the number General John Nicholson, the top US military commander in the country, requested.
Criticising previous administrations, he said: "We will not talk about numbers of troops or our plans."
However, Mr Trump said there would be an escalation in the battle against groups like al-Qaeda and so-called Islamic State.
"[They] need to know they have nowhere to hide - that no place is beyond the reach of American arms," he said.
But he also indicated there could one day be a peace deal with the Taliban.
"Some day, after an effective military effort, perhaps it will be possible to have a political sentiment that includes elements of the Taliban in Afghanistan," he said.
"But nobody knows if or when that will ever happen."
US combat operations against the Taliban officially ended in 2014, but special forces have continued to provide support to Afghan troops.
Current US troop numbers in Afghanistan are about 8,400.
The Afghan government continues to battle insurgency groups and controls just half of the country.