Many people are under the impression that you need to find skill gems from monsters or chests in Path of Exile, but as a matter of fact, most of them can be bought from NPCs in the game. In order to be able to buy them you need to have completed a certain quest, but once you have completed all quests in normal difficulty, you can buy all the skill gems you need.
Some gems are only sold by certain classes, so you cannot buy every single gem from a single character. To get the other gems you either need to level up another class, or buy the gem from another person. An easy tool to use for this is poe.trade. Just search for the gem you need, and whisper the player to buy it.
To buy skill gems, go to any major hub and purchase items from the vendors there. Some of them will have a second page of items for sale, and this is where the skill gems are located. Note that the supply is different in all acts, so make sure to check out all the different hubs to find all the skill gems you can buy.
The skill gems are typically very cheap, only costing some low-end currency like Orb of Alteration.