Before travel to another country it is required to check whether you need to obtain a visa to visit this country. If you are lucky owner of an American passport, you are able to visit most of countries without any visas. But for example if you want to visit Russia, you'll have to obtain Russian visa in advance. Without it, you won't get into the country. Moreover, without Russian visa you won't even be admitted to a plane that flies to Russia.
Citizens of many other countries have to get visas almost constantly. However, it's usually not very complicated to obtain a visa. All information and a list of required documents usually can be found at Embassy website of the country where you are going to visit. I strongly recommend not to use the services of travel agents. In reality, their assistance often do more harm than benefit.
Before you go to get a visa, check whether there are any visa exemptions for your case. For example, citizens of most countries require a visa to visit Singapore. But they have the right for a non-visa transit via Singapore, when fly from one country, and next fly out to another country, if stay in Singapore does not exceed 96 hours. Obviously, it's quite enough to explore this beautiful city in 4 days.
Another specific situation is a transit through the country without leaving the airport. For example, many (but not all) of major airports in Europe have a transit zone, where you will wait for the next flight. Without a visa you won't get outside the airport, but you will be able to fly to your next destionation in other country. Visa-free transit has many nuances, so it is better to study this issue in advance and print authorized materials from the Internet. The problem is that airline staff often refuse to check-in those who don't have a requred visa regardless visa-free transit. In this case, relevant printouts can help you greatly.
And the last. There is no non-visa transit in the USA. Even if you arrive and immediately fly out of the country from the same airport, you still need to obtain a visa! Of course this does not apply to citizens of those countries who don't require a visa to visit the United States.