To unlock all the achievements related to stars, you need to collect a total of 200 of them. There are 3 on each level and it is not that hard to collect most of them. You will need to collect at least 300 stars to be able to unlock the “Robo Master” achievement (need to unlock Level 6-23), so this will come naturally. You can check your total in the World-Selection screen. You need to finish a level (by getting the candy in the creature's mouth, regardless of collecting any stars) or skip the level (pausing and hitting "Skip") to unlock the next level of the world. You unlock new worlds (the first level of each) by gathering a certain number of stars.
See THIS link for 3-star video walkthroughs for each level of the game if you are having trouble. Thanks to cuttheropecheats.com for posting these videos. It is recommended that you only use the walkthrough if you get stuck on a level or need to pick up a few extra stars later in the game, as the levels are mostly very easy and a guide will just slow you down in most cases.