It is something like this. Your overall level up, that light purple bar fills up automatically when other skills in the constellation tree level up. And those skills level up only when you use them frequently. If you don't, they wouldn't budge, not even a bit. So if you are a warrior specializing in combat skills, your magic skills like alteration, conjuration, destruction etc or stealth skills like alchemy, sneak, speech etc wouldn't improve and vice-versa.
Each skill in the constellation tree has its associated perks which you need to unlock as you level up. Do note that these perks when unlocked remain permanently unlocked for the rest of your progress, cannot be undone and reassigned later.
Some NPCs can help you train a particular skill (you'll come across them eventually as you explore Skyrim) but is limited to 5 levels per overall level. And they charge as you well. This charge increases phenomenally as you seek training from Master/Expert trainers. Some skill books also help you increase the skill by 1 level. Sometimes, completion of a quest rewards you with a skill level-up by that NPC who gave you the quest.