"It's a real team, and we want to do it better than ever before," Trump said during a meeting with officials from local, state and federal agencies.
Trump, who arrived wearing a white USA baseball cap, was briefed on relief efforts in Corpus Christi before touring an emergency operations centre in Austin.
He received another briefing with state leaders.
Senators Ted Cruz and John Cornyn were on hand, as was Representative Blake Farenthold, whose district encompasses Corpus Christi.
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Texas Governor Abbott briefed Trump on the situation at a convention centre in downtown Houston, where 9,000 residents fleeing rising floodwaters have crammed into a makeshift shelter designed to accommodate 5,000.
"At the convention centre, we are sustaining food," said Abbott. "All eyes are on Houston, and so are mine," he added.
Trump said: "We want todo it better than ever before. "We want to be looked atin fiveyears, in 10years from now as,this is the way to do it. This [storm] was of epic proportion.
After conferring with emergency management officials, the president, accompanied by first lady Melania Trump, plans to travel north to Austin, where he will meet other officials.
The president is also scheduled to meet with several members of Congress, and local elected officials and mayors.
"The president wants to be very cautious about making sure that any activity doesn't disrupt any of the recovery efforts that are still ongoing, which is the reason for the locations we are going here today," she said.
Trump, who pushed aides to schedule a visit to Texas as early as possible after Harvey made landfall near Rockport on Friday night as a Category 4 hurricane, initially considered touring San Antonio, which is outside the most hard-hit areas.
But he settled on Corpus Christi because it was 30 miles away from the most severely impacted parts of the Gulf Coast, and suffered relatively light damage from the initial impact of the storm.
At least 10 people have died in storm-related incidents. More than 30 inches of rain has swamped parts of Houston, and more is likely, potentially reaching 50 inches until Thursday.
Vice President Mike Pence, who did interviews with two Texas radio stations today, framed Trump's eagerness in the context of the president being "a leader that is absolutely committed to making sure things are happening the way they're supposed to be happening.
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