Since we crochet back and forth naturally, the easiest way to add color to a project is with horizontal stripes. Adding a new color via striping (introducing new yarn) is done by dropping the yarn in use at the end of a row, and adding the new color at the start of the following row. This can be accomplished by:
a) fastening off the first color and pulling up a loop of the next color in the first stitch, or
b) working the last stitch almost to completion, then pulling up the last loop with the new color to be used (regardless if working sc, dc, trc, etc.).
The above is the simplest way to "pop up" your color palate and should not be too intimidating for a beginner.
Of course, the question of "to knot or not" comes up. So here is my answer. DO WHAT SUITS YOU, and I won't judge you. If you are comfortable weaving cut yarn tails into their own colors, do so. If you prefer to knot 2 tails together before weaving into their own colors, do so. Now I know some will cringe at the idea of tying knots...but here is the thing: sometimes yarn is knotted inside a skein, right from the manufacturer...so it happens. Secondly, a well hidden knot is nobody's business - :) Lastly, if a knot brings a bit of comfort to a crafter that their gift won't come undone in the wash...I say do it. Tie a knot. Matter of fact, some people make a piece that is all knots...scraps tied together and crafted into...whatever, and the knots are a featured part of the piece. Whatever gets made from that mystery ball of knotted yarn is certainly UNIQUE~!
WHEN TO NOT TIE A KNOT: Ah, you knew this was coming, didn't you........here is when:
There are other ways to add color, such as blocks of color, vertical stripes, or as in a charted design, by carrying, weaving or crossing the yarn. For these color techniques, the use of separate skeins or bobbins simultaneously may be necessary, depending on how many colors are in use. For some colorwork, checkerboard for instance, the yarn may be woven into the piece by carrying it across the stitches of one color block and crocheting over it....then dropping the color in use and picking up the color that has just been woven over, etc. Crocheting over a “not in use” color is similar to hiding yarn tails at the beginning of a row. Literally, you crochet right over the yarn to be carried to the next space, and it disappears into the stitches. This could cause a problem, however, if the yarns in use are terribly opposite to each other...in those instance, you may be able to see the yarn carried over...it is doesn't bother you; then it isn't a problem...it becomes the nature of the piece. :)
An alternate method, carrying the yarn from section to section along the "wrong" side of a project, may be used for items where only the front of the piece (for instance, a chair cover or wall hanging) will be seen. In carrying yarn across the backside, if it spans more than 4 stitches, the yarn should be caught into the stitches being worked every 5 stitches or so. To to this, at intervals of 5 stitches, catch the stranded yarn with your hook and work over it, that is, crochet over it with that one stitch, then strand it again for 5 more stitches, etc. and so on. The reason for this is that since it is one-sided, you don't want the bulkiness that working over each stand will bring, and also perhaps it is a very dark color in contrast to the main yarn, and you would not want that to show through to the front. (For knitters, this is quite similar to fair isle/stranding technique.)
Lastly, if the design is suitable to the need for various bobbins, with lots of color blocking through the design, each color should be worked to its necessary point, then dropped and said yarn crossed with the next color needed, and so on. How this is done is simple. Drop the bobbin containing the current yarn and bring the next color yarn bobbin up and under to the right of the bobbin dropped, which twists the 2 yarns, then proceed with the next stitch, pulling it a little bit tighter than usual. Proceed throughout the piece this way at each color change.
We make many decisions based on the project itself. This is no different crocheting with color. The design will indicate which method is best.
Basically, color changes without cutting yarn are made by working the LAST loop of the last stitch of one color with the next color.
If you are tying knots at the end, make them as inconspicuous as possible and weave each tail into its own color.
When using bobbins, it is important to keep them untangled. Seriously.
When carrying yarn, don't carry it across too many stitches; remember to catch it every 4 to 5 stitches.
VERY IMPORTANT: Carry the yarn loosely across the back so that it doesn't cause your item to pucker...but not so loose that your beginning and ending stitches are obviously loose.
Yarn tails throughout a spectularly colored piece should always be woven into a yarn row of the same color.
When using the crossed yarn method, use the necessary amount of bobbins for each separate section of color. It is worth the time, trust me.
Sometimes, even when using the carrying method, it may be best to utilize separate bobbins for different sections of the same color to avoid carrying yarn across wide expanses.
The above is my experience and opinion. If anyone has something to add, your comments, tips or tricks are most welcome and invited. Again, look to the internet for videos and\or tutorials.