Guitar picking, similar to guitar playing, can be as simple or complex as you want to make it. Most people are content with the basic up/ down picking patterns whereas the more you delve into tone, feel and approach, you may find you'll get more mileage out of using various picking techniques to add variety and diversity to your playing. For this lesson, I'll be talking primarily about single note picking as opposed to chord strumming.
Let's begin by discussing the basics: what type of pick to use and how to correctly hold a pick. The type of pick you use is personal preference and is usually dictated by the style of music you prefer to play. The heavier the style, the heavier the pick. Because I play various styles, I like using medium guaged picks - they have some flexibility but are also sturdy enough to withstand heavier playing as well. I also like to use small guitar picks as I feel I have more control over them than larger sized guitar picks.
Regarding picking technique, there's really no "right" or "wrong" way to hold a pick - choose whatever feels most comfortable to you. Most players choose to hold the pick in relation to the strings similar to photo 1 with the flat side down whereas I hold my pick at an angle as depicted in photo 2. The first example allows the player to play at consistent levels and consistent tones. The second example, in my opinion, allows the player to change tones and play at various volumes dependent upon their picking attack. Although you may sacrifice some speed when using this approach, it offers more variety and control over your playing.
Photo 3 shows an alternate way of holding your pick when finger picking instead of discarding it completely. Experiment and use what works best for you.
Another approach to consider is if a pick is necessary at all. By using your fingers, you'll create a warmer, fatter sound which will drastically change your tone. In Example 1 and Example 2, you'll hear the same musical passage but each are played differently - take a listen to hear the difference between playing with a pick and playing with your fingers. Example 3 takes that concept a step further and combines the two techniques to create what's known as "hybrid picking" in which you use both your pick and fingers (often referred to as "chicken picking" in country music). Players such as Wes Montgomery and Jeff Beck are excellent studies in those who prefer to use their fingers as opposed to a pick while country legends Albert Lee and Chet Atkins are considered to be two of the premier "chicken picking" guitarists to ever pick up the instrument.
One last point to mention is also the pattern your picking pattern may dictate the style of music you will be playing. For example, many punk and metal guitarists will predominately use downstrokes to play their heavier riffs whereas reggae players will use upstrokes as a result of the rhythms falling on the upbeat. Of course there will also be exceptions to the rule (if there is such a rule) but these are just simple suggestions to consider and begin to add to your style.
I hope these different suggestions and playing examples open some doors for you to explore. Using these various techniques will help bring some interesting elements to your playing and songwriting. Don't be afraid to apply them right away in the hope of bring some new ideas to your playing. Best of luck.