Africa

In January, it is winter in North Africa; on the west coast the sea is a bit cold, down to the latitude of central and northern Senegal. On the coast of the Gulf of Guinea it's hot and there is little rain, but the sky is often cloudy in the easternmost part. In Eastern African countries near the Equator (Kenya, northern Tanzania) there is little rain and the sun shines, but it's also hot and muggy.
Where to go: Benin, Cote d'Ivoire, Ghana, Djibouti, Guinea, Guinea Bissau, Kenya, Tanzania (north), Togo, Zanzibar

Asia

It is winter, and in Southern Asia the north-east monsoon prevails, which brings fine weather, except in the more exposed slopes.
Where to go: Burma, Philippines (western side), India, Maldives, Malaysia (western side, see Langkawi), Oman (south), Thailand (Koh Samui, Pattaya, Phuket), Vietnam (south), Yemen

North and Central America

In Florida, it can sometimes be cold, especially in the north-central part; in Yucatan, cloudy days, even cool and windy, are possible, with some rain. In Central America it is hot, but the slopes exposed to the northeast trade winds are rainy.
Where to go: Costa Rica (Pacific coast), El Salvador, Guatemala (Pacific coast), Honduras (Pacific coast), Mexico (Acapulco), Nicaragua (Pacific coast), Panama (south side)

Caribbean Sea

It's generally a good month, but in the northernmost part, there may be windy and cloudy days when the cool wind called El Norte blows from the United States. In the northern slopes of the larger islands and in the Windward Islands, the trade winds can bring some showers. The warmest and sunniest islands are therefore the southern ones (Leeward Antilles), where, however, showers are quite common, although usually short-lived.
Where to go: Anguilla, Antigua and Barbuda, Southern Bahamas (Inagua and Great Inagua), Barbados, British Virgin Islands, Colombia (Caribbean Sea), Dominican Republic (south and east coast), Grenada, Guadeloupe, Jamaica, Puerto Rico (south and west coast), Saint Kitts and Nevis, Saint Barthélemy, Saint Martin/Sint Maarten, Trinidad and Tobago, Turks and Caicos, U.S. Virgin Islands, Venezuela (Maracaibo)

South America

It's summer in the Southern Hemisphere, which means that there are widespread rains at tropical latitudes. Further south, the sea is quite cold both on the east coast, in Argentina and Uruguay, and on the west coast, up to Chile.
Where to go: Brazil (Rio de Janeiro, Salvador), Peru

Oceania

It's summer in the Southern Hemisphere, therefore at tropical latitudes it's hot, it rains and tropical cyclones can pass. In Australia, in Perth it's sunny, and the sea temperature is a bit cool, but overall acceptable.
Where to go: Australia (Perth), Marshall Islands, Northern Mariana Islands, Micronesia