Convicted felons are able to obtain U.S. passports under most circumstances. If a country requires only a passport for admittance, gaining entry as a tourist is normally not a problem with some exceptions. Obtaining a visa, however, can be more difficult.
Many countries require a visa either prior to travel or upon arrival for entry. Many visa applications inquire about your criminal record. Countries which ask about your criminal past on visa applications include: China, Japan, Australia and Fiji. If you have a felony conviction, you may be denied entry to these countries. Note that U.S. citizens must obtain pre-arrival visas for travel to India and Brazil, but their applications do not inquire about criminality; therefore, a felon with a valid U.S. passport should not have a problem travelling to either country.
Dozens of countries do not require visas for U.S. citizens. Thus, if you have a valid U.S. passport, entry should be straightforward and no different than as if you had no criminal record. Some foreign nations which do not require visas for U.S. citizens include: Mexico, all Caribbean countries, all countries in the European Union, Colombia, Venezuela, Peru, Ecuador, Thailand, New Zealand and South Africa.
Travel to Canada may be possible if you are a convicted felon under special circumstances. According to the U.S. State Department, anyone with a criminal record--including misdemeanours or Driving While Impaired (DWI)--may be barred from entering Canada and must obtain a special waiver in advance of any travel. Visit the Canadian citizenship and immigration website or the Canadian Embassy for information and to apply for a waiver.