Multiple methods are possible. You can even have a certain "style".
1) has some good examples above, structure of the prime implied is really good. Finding a counter example in your head, another good approach.
2) As mentioned elsewhere, if you want to prove something isn't prime, you want to find something that divides it. If you are given a list of integers, you often want to look at its remainders with respect to some other integer.
Putting together a toolbox of Rules of Thumb is very helpful in such problems.
One Rule of thumb, every prime over 5 is one less or more than a multiple of 6. If you can prove that one of n,n+2, or n+4 is not one more or less than a multiple of 6, then you are done. You can go in order for that. n has to be one less than or more than a multiple of six are the triple wont work, n is composite. If it's one less than a multiple of 6, then adding 4 will make it 3 plus a multiple of 6, so n+4 will be divisible by 3. If it's one more than a multiple of 6, then n+2 will be divisible by 3.
5xum's answer is nice. Every number is either a multiple of 3, or one more or less than a multiple of 3. If n is a a multiple of 3, the theorem is proved. If n is one less than a multiple of 3 then n=3k-1 for some integer k. Then n+4=3(k+1). So you get a multiple of 3. If n is one more than a multiple of 3, then n=3k+1 for some integer k. So n+2=3(k+1). So no matter what your n, one of n, n+2, and n+4 is divisible by 3 and therefor not prime.
You can take a similar approach here: How do you know that n(n+1)(2n+1) is always divisible by 6? Well, if n is even then the product is too. If n is odd, then n+1 must be even, and so is the product. So we know our product is divisible by 2. Now can you see how n(n+1)(n+2) must be divisible by 3? Let's suppose neither n or n+1 is divisible by 3. Then n+2 is. Then so is 2n+4. Take 3 away and you have 2n+1. Three less than a multiple of three must be a multiple of 3, so 2n+1 is divisible by 3. The product is divisible by 2 and 3, so it must be divisible by 6.
Rules of thumb help develop intuition, a very important quality in a mathematician. For example out of n,n+2, and n+4. Suppose n and n+2 are both prime. This happens, the pairs when this happens are called Twin Primes. No one knows if there are an infinite number of Twin Primes. You never hear of triplet primes. Is that just a coincidence? Intuition probably tells you no and indicates an opportunity to explore the issue.
I had a math teacher in high school. He said "If you don't know 5 ways of solving a problem then you didn't understand it well enough." There's no wrong way to prove a theorem. You can get very creative with it. Mathematics offers much more latitude than many would have you believe.
Keep working problems. Keep being curious. Develop intuition by building that tool box. Follow your curiosity. Figure out ways to tweak your problems so that the techniques you have used before no longer apply.