By Liza Lin

Rules that have slowed the new coronavirus were falling away in some U.S. states, Italy and other countries across the world this week as drugmakers raced to develop a new vaccine and China pledged $2 billion to fight the pandemic beyond its borders.

Italy lifted many of Europe's earliest and strictest virus-containment curbs on Monday, allowing restaurants, cafes, clothing retailers, hairdressers and museums to open their doors for the first time in two months.

Shops and small businesses also welcomed back customers in Belgium, the Netherlands, Spain and Lebanon. Junior high-school students were back in class in parts of France. Beaches in Greece and some shopping malls in Thailand reopened as governments concluded that declining infection rates allowed for a significant relaxing of restrictions on how people move about, despite the risk of a resurgence.

In Texas, gyms and movie theaters were set to reopen on Monday even as infections and virus-related deaths there were reported on the rise. Idaho on Saturday allowed dine-in restaurants to resume operations under physical-distancing rules. New Jersey permitted chartered-boat services and watercraft rentals to resume on Sunday.

Warmer weather drew large crowds to beaches in New York City and New Jersey over the weekend, highlighting the challenges that the hardest-hit parts of the country face in enforcing distancing rules this summer. New York Mayor Bill de Blasio said he doesn't want the city's beaches to open for swimming by Memorial Day weekend.

New York Gov. Andrew Cuomo on Sunday urged New Yorkers to get tested for Covid-19. Mr. Cuomo said the state is conducting about 40,000 Covid-19 tests a day at more than 700 sites. New York has more than 350,000 infections and 22,600 confirmed virus-related deaths, according to figures from the state.

More than 4.7 million people across the world have caught the coronavirus and more than 315,000 have died of Covid-19, the respiratory disease it causes, according to data compiled by Johns Hopkins University. Nearly 1.5 million cases of infection and more than 89,500 deaths have been recorded in the U.S., the nation worst affected, according to the data.

Experts caution that official numbers likely understate the extent of the pandemic.

Public health officials say a safe and effective vaccine is the best way to prevent the disease and curb its transmission. Drugmakers are considering how to roll out vaccines, with the first batches reserved for front-line medical workers. About 100 vaccines are being developed globally, and eight of those -- from companies like Moderna Inc. and Pfizer Inc. -- are already being tested on humans.

Chinese President Xi Jinping said his government would spend $2 billion fighting the coronavirus around the world, particularly in Africa. He called on other countries to fund the World Health Organization, offering his country as a global leader prepared to support a United Nations agency from which the Trump administration has withdrawn funding.

Mr. Xi spoke Monday at WHO's annual summit, the World Health Assembly. Convened via videoconference, representatives of 194 nations were set to debate lessons learned since the virus was first detected in late December in China's Hubei province. The European Union and Australia brought a resolution backed by more than 100 countries calling for an independent review of how governments responded to the pandemic and whether WHO acted to the best of its abilities to contain the disease.

In Asia, where infection rates rose early in many places, China and other countries took the lead in closing their societies and then in loosening restrictions.

South Korea reported 15 new cases on Monday and one new death. In a reflection of the country's downward trend on fresh infections, the U.S. military in South Korea eased restrictions against dining out, going to off-base gyms, shopping and other outdoor activities.

Samsung Electronics Co. Vice Chairman Lee Jae-yong visited one of the South Korean company's semiconductor factories in Xi'an, China, on Monday -- a visit the company said was the first by a global business leader to China since the coronavirus outbreak became a global health crisis.

In Thailand, one of the country's largest retail-property chains opened its malls after nearly two months. The company, Central Pattana PLC, said it opened its 33 malls nationwide to shoppers on Sunday, while introducing new measures to space out customers and facilitate contact tracing. Those include scanning a QR code upon entering and leaving, in addition to providing contact information, the company said.

Australia's most populous state on began its first full week out of lockdown. Authorities in New South Wales released a plan to help reduce citizens' use of public transportation during peak hours. The state, home to the city of Sydney, will establish temporary parking lots and bike lanes and increase access to pedestrian paths.

Italy's emergence from two months of severe limitations was visible in Rome, with the return of traffic to major roads and a surge of bicycle riders. Priests resumed celebrating Mass.

But it's not back to normal. Schools are shut until the end of the school year. Cinemas and theaters are closed at least until mid-June. Travel between regions is still largely banned. And as in other countries, the shops and other public places that are reopening have to be sanitized regularly and distancing rules must be enforced.

Some business owners, including Arrigo Cirpiani, the owner of Harry's Bar in Venice, say the rules are so strict they may not reopen for now. Under the new rules, the bar could only fit around 10 customers, down from the usual 150, he said.

Italy, one of the countries hardest hit by the pandemic, has recorded declining infection rates for weeks but the virus lingers.

Other countries that have eased freedom of movement are keeping other restrictions in place.

"Countries around the world are going toward coexistence with the virus, " Egypt's Prime Minister Mostafa Madbouly said Sunday as he ordered people to wear masks in public places, public buildings, trains and buses.

Some governments were struggling with new outbreaks.

Since easing its virus-fighting restrictions this month, France has discovered 25 new infection clusters, and is attempting to contain them by isolating those involved and testing people with whom they came in contact, France's health minister said.

India on Monday registered its highest single-day increase in confirmed infections for the second day in a row. The Ministry of Health and Family Welfare reported more than 5,200 new cases, bringing India's total to 96,169. More than 3,000 people there have died from the disease, the ministry said.

The numbers offered a fresh challenge to authorities, who had sought to ease lockdown restrictions. India's government on Sunday extended its nationwide lockdown to the end of the month.

In China, fears about a new wave of cases were sparked this month after new clusters were discovered in the northeastern province of Jilin.

China reported seven new cases as of the end of Sunday, its National Health Commission said. Four were imported, while two of the three domestic cases were discovered in Jilin, which borders Russia and North Korea.

Chinese Vice Premier Sun Chunlan traveled to the province over the weekend to learn more about efforts to control the virus, according to local media. Authorities fired six senior provincial officials on Saturday, according to the province's account on the WeChat messaging service.

Japan on Monday said the pandemic helped push the country into a recession. The world's third-largest economy after the U.S. and China shrank by an annualized 3.4% in the first three months of 2020. That followed a contraction of 7.3% in the previous quarter due to an increase in the country's sales tax. Two straight quarters of contraction is one common definition of a recession. Japanese Economy Minister Yasutoshi Nishimura said a worse result is expected in the current quarter, after Prime Minister Shinzo Abe declared a national state of emergency in April.

Write to Liza Lin at Liza.Lin@wsj.com