Sen. Dianne Feinstein of California, a veteran Democrat who was sworn in to her sixth term Thursday, stopped in her tracks when she spotted freshman Sen. Marsha Blackburn, Tennessee Republican, in the Capitol basement.

“I look forward to a good year,” Ms. Feinstein said, as they shook hands.

“I do, too,” Mrs. Blackburn said.

The Senate was an eye of calm amid a storm brewing between House Democrats and the Trump White House — the main combatants in a reshaped Washington that rang in a new Congress on Thursday.

Unlike the House, where power shifted dramatically to the left, Senate Republicans expanded their majority from 51 seats to 53 in the midterm cycle, leaving GOP leaders to hope they can forge cooperation with Democrats.

“We know that the Senate, with this Republican majority, is fertile soil for big bipartisan accomplishments,” said Majority Leader Mitch McConnell. “The question is: Will the newly Democratic House join in this good momentum, or bring it to a standstill?”

While the House had a full day of action, including late-night votes on spending bills, the Senate called it a day after a few hours of pomp.

Vice President Mike Pence delivered the oath of office to returning incumbents, six freshmen Republicans and a pair of new Democrats, Sen. Kyrsten Sinema of Arizona and Sen. Jacky Rosen of Nevada, who reflect the increasingly blue tint of states bordering Democratic stronghold California.

Sen.-elect Rick Scott, Florida Republican, will be sworn in next week, after he wraps up his duties as the state’s governor.

The Senate also elected a new president pro tempore in Sen. Chuck Grassley of Iowa, the chamber’s senior Republican, who takes his place as third in the line of presidential succession, behind the vice president and House speaker.

He was sworn in by Vice President Mike Pence, who also swore in groups of senators on the floor, before retiring to the Old Senate Chamber to recreate the scene in ceremonial photo-ops attended by senators’ family and friends.

Most senators showed off their grandchildren to Mr. Pence. Ms. Sinema, though, showed up solo for her photo-op, and after looking at the tape marks on the floor showing where families were to stand for the photos, she quipped, “Could we get a spouse? Just kidding.”

Ms. Sinema will be one of the most-watched freshman, coming from a GOP-leaning state. In a video release Thursday she promised to seek out deals.

“I promise you this, I will continue to work with anyone to get stuff done for everyday Arizonans,” Ms. Sinema said. “You all deserve leaders who put country above party, and I pledge to keep our state’s tradition of doing just that.”

Republican freshmen who ousted Democrats in November said they have no intention of being potted plants amid skirmishes between the liberal-minded House and Mr. Trump.

Sen. Mike Braun, Indiana Republican, said he wants to make sure budgets are written and approved. He wants to address health care and infrastructure, the latter of which Democrats have pointed to as an area of potential compromise.

Sen. Kevin Cramer, North Dakota Republican, highlighted the Senate’s vital role in the “personnel business,” approving or rejecting Mr. Trump nominees to key jobs or judgeships.

“The advise and consent role for appointments, particularly for the judiciary, which has such long-term implications, is the most obvious,” he said.

One early test for all members will come on the partial government shutdown, which will reach two weeks on Friday.

Though voters increased the GOP Senate majority, they will still need Democratic votes to overcome a filibuster. And even if bills get 60 votes, both parties need to get President Trump to sign whatever they produce.

Democrats said the best way to get off on the right foot is to reopen the government and fight over Mr. Trump’s border demands at a later date.

“If people are willing to punish folks unnecessarily, that will set a bad tone,” said Sen. Tim Kaine, Virginia Democrat. “If they’re willing to focus the dispute on the area in dispute, it will be a positive.”