Africa

In June, in the Northern Hemisphere the summer begins, and it starts to get hot also on the Mediterranean coast. The Sahara is by now on fire. The African monsoon continues its advance to the north, affecting in full force the countries of the Gulf of Guinea. In the Southern Hemisphere winter begins, and even at tropical latitudes it can get a little cool. You can visit the cities that are located around the Equator or a little more to the south, and at high altitudes.
Where to go: Brazzaville, Dar es Salaam, Harare, Kampala, Kigali, Libreville, Lilongwe, Luanda, Lusaka, Maputo, Mbabane, Nairobi

Asia

In June, summer begins in the temperate and in the northern zone, and in many continental areas it's too hot; by now the monsoon has reached most of Southern and Eastern Asia, but the dry season begins in some parts of Indonesia.
Where to go: Ankara, Jakarta, Kuala Lumpur, Lhasa, Omsk, Tbilisi (first half), Ulaanbaatar

North and Central America

It's the first summer month. The best season begins in the northern lands. It's often too hot in the continental zones of the United States, and sometimes even in Southern Canada. The heat is sweltering in the tierras calientes of Central America. The hurricane season starts in the Gulf of Mexico, and also in the Pacific Coast of Central America, however, they will be more likely in the following months.
Where to go: Anchorage, Ilulissat, Montréal, San Francisco, Seattle, Toronto, Vancouver

South America

In June, winter begins in the Southern Hemisphere: at sub-tropical latitudes it's cool or cold. On the Pacific Coast, fog and low clouds are common. In the north of the continent, in the Northern Hemisphere, it is summer, so it's hot and rainy. The dry season begins in the Peruvian Andes. You can go to cities that are not located too much to the south, where winter is mild.
Where to go: Asunción, La Paz, Rio de Janeiro, San Paolo

Europe

June can be considered as the first summer month, typically the best season to visit nordic countries. On the 21st of the month it's the summer solstice, the longest day of the year: in cities of Northern Europe you can see the white nights for the entire month. Compared with July and August, June is cooler, but in Western and Northern Europe, it is often the driest month as well. In Southern Europe, it begins to get a little too hot, or at least some heat waves from Africa are possible.
Where to go: Amsterdam, Barcelona (first half), Belgrade, Berlin, Bratislava, Brussels, Bucharest, Budapest, Chisinau, Copenhagen, Dublin, Edinburgh, Geneva, Helsinki, Istanbul, Kiev, Lisbon, London, Luxembourg, Milan, Minsk, Moscow, Oslo, Paris, Podgorica (first half), Prague, Reykjavik, Riga, Rome (first half), St. Petersburg, Sarajevo, Skopje (first half), Sofia, Stockholm, Tallinn, Tirana, Tórshavn, Warsaw, Vienna, Vilnius, Zagreb

Oceania

In June, winter begins in the Southern Hemisphere, which means that the weather is good in the islands that are located at tropical latitudes (but not too far to the south), where you will find sunshine, little rain, a pleasantly warm weather and tempered by the breeze. In Northern Australia, it's the dry season, so it's the best time, even though it's hot.
Where to go: Brisbane, Darwin, Honolulu, Papeete, Port Moresby