Democratic prosecutors sparred with Mr Trump’s lawyers over the process, while Republicans rejected their demands for more witnesses.
The trial will resume on Wednesday with arguments by the prosecution, to be followed by the defence and questions.
Mr Trump is the third US president to undergo an impeachment trial.
He is charged with abuse of power and obstructing the congressional inquiry. He denies wrongdoing.
Mr Trump is being put on trial after being impeached last month by the Democratic-led House of Representatives.
But the Senate, which is controlled by his fellow Republicans, is not expected to convict and remove him from office.
At the World Economic Forum in Davos, Switzerland, on Tuesday, Mr Trump dismissed the accusations against him as “just a hoax”.
What happens now?
Senators have taken oaths to act as impartial jurors in a trial presided over by the US Chief Justice, John Roberts. House Democrats known as “impeachment managers” act as the prosecution, while Mr Trump’s legal team acts as the defence.
Under the rules agreed on a first day of proceedings that finished close to 02:00 local time (07:00 GMT) on Wednesday, each side will be given up to 24 hours to lay out their case in opening arguments, over three days.
This will begin on Wednesday afternoon. After this finishes, probably early next week, senators will have a chance to ask questions. They have been given 16 hours.
Then attention will return again to the key issue of new witnesses and evidence.
Democrats want to hear from key White House aides who worked closely with Mr Trump, including acting Chief of Staff Mick Mulvaney and former National Security Adviser John Bolton.
But most Republicans are loath to let this happen.
How were Democrats blocked on Tuesday?
By party-line votes of 53-47, the Senate rejected three Democratic bids to obtain documents and evidence in the impeachment trial.
Senators blocked a motion from Democratic leader Chuck Schumer to subpoena White House files related to Mr Trump’s dealings with Ukraine.
They also rejected follow-up motions demanding a subpoena of records and documents from the state department and White House budget office.
Republican Senators also turned back an effort by Democrats to subpoena Mr Bolton, who has said he would comply with any such order.
In his opening statement, Adam Schiff, the House Democrat leading the impeachment case, said most Americans “do not believe there will be a fair trial”.
“They don’t believe the Senate will be impartial,” he added. “They believe the result is pre-cooked.”
The president’s legal team had earlier demanded he be immediately acquitted, calling the trial “a dangerous perversion of the constitution”.
At one point during bitter arguments, Justice Roberts admonished both the House prosecutors and the Trump legal team, asking them to remember that they were “addressing the world’s greatest deliberative body”.
How did Mitch McConnell come under pressure?
Backed by the president’s lawyers, Senate Majority Leader Mitch McConnell had initially planned to condense the opening arguments from three days to two.
Democrats said this would have been no less than a cover-up.
But after a meeting with senators, including some Republicans, Mr McConnell agreed on Tuesday to three days for opening arguments.
The senators had expressed concern about how middle-of-the-night sessions would look to US voters.
White House Counsel Pat Cipollone, the president’s lead lawyer, said: “It’s a fair process. There is absolutely no case.”
Several more days of procedural tangles are expected.
What are the charges?
First, the president is accused of seeking help from Ukraine’s government to help himself get re-elected in November.
It is claimed that, during a call with Ukrainian President Volodymyr Zelensky, he held back military aid as he sought an anti-corruption investigation into Democratic White House candidate Joe Biden, whose son, Hunter, held a board position with a Ukrainian energy firm, Burisma.
The second allegation is that, by refusing to allow White House staff to testify at the impeachment hearings last year, Mr Trump obstructed Congress.
The Senate is hearing the case as the Democratic-led House voted to impeach Mr Trump on 18 December.