These achievements will unlock as you progress through the single-player game. There are 5 worlds which unlock by earning stars on all of the levels on the previous worlds (first world is unlocked automatically). The first level of the world will unlock when the world does. Progression through the levels of a world is done by earning at least one star on each level, which will unlock the subsequent level. Stars are only based on your total score and the their requirements are shown when you first click on a level to play.
When you unlock the Desert and Tundra worlds, you must actually go in the world selection screen and click on the respective world for the achievement to pop. The last world, Coastline, unlocks when you earn 99 stars (out of 180 on the first 4 worlds). To unlock "Everlasting Globetrotter," you need to progress all the way to level 15 and the achievement will unlock when you complete it (you do not have to get any stars).