I often hear beginners ask "How do I change color?" or "how do I add a new ball when the first ball runs out?" These are great questions! Those of us who are veteran crocheters don't even think about it, but everyone has to learn somewhere! So why not here?
Whether you are changing color or adding a new ball of the same color, it's the same technique. Always stop the stitch one step short of completion, yarn over with the new yarn and finish the stitch, continue working with the new color. I'll show you below. When working in rows, always try to change color on the last stitch of the row, so that the ends can be woven in neatly along the edging. When making a garment, try to change color under the arm or on the side seam. When working in rounds, like for motifs, change color when you start a new round.
Sometimes when there is a flaw in the yarn, like a knot where it was broken during manufacturing and cut and tied really short, or when there is a bumble, a fuzzy lump in the yarn, you need to cut the yarn before the flaw, cut after the flaw so that you are cutting the flaw or knot out completely, then pull the new yarn through as shown above so that the flaw is not in your project. Those tiny knots made by the manufacturer pull undone very easily and you don't want that to happen in your finished project! Also, the fuzzy flaws represent a weakness in the yarn in addition to an annoying change in texture.
Above, to add the pink, the instructions might say, "join pink with sl st in 3rd dc of dc-5 side", so, with no slip knot, I'm choosing to just insert my hook into the stitch, and leaving a 4" tail or so, pull the new yarn, slipping it through the st, (that's the slip stitch), then continue with the instructions, usually ch 3 or something. If it says to join with a single crochet, I do the same thing, I pull the yarn through, ch 1, then sc. I think others do it differently. You can put a slip knot on the hook if you want, but the knot adds a little bit of bulk, and since the end will be woven in anyway, I don't feel like it's necessary.
Questions about changing yarn? Let me know and I'll try to generate more photos to answer.