When not to look at p-value I am trying to indentify correlations between the columns of a dataset I have. I am using the pearson correlation index, and am looking at the p-value I get from the "correlation test" of R. I know that, in theory, the p-value is the probability of the correlation being due to chance. Now, what I wanted to know, can I always look at this value to have a good metric of how certain the relation between the variables is, or are there any assumptions that have to be statisfied?
EDIT
I'll add some context in order to better explain the question
I am actually studying the relationship between some variables I extracted from a tv series (eg number of character, number of cuts, IMDB rating etc.) and wanted to see if there is any kind of relation between them. In order to do this, I thought correlation might be a nice idea. Some variables have got interesting correlations, with very little p-values, and reading online it seemed like a good idea looking at this value.
After the comments I am not sure I really understood what this value means, but reading online I found this:
"The p-value is the probability that the null hypothesis is true. In our case, it represents the probability that the correlation between x and y in the sample data occurred by chance" (source).

If you are looking for correlations, your first look should not be at the p-value, but at the value of the correlation coefficient $r$. High correlation means high $|r|$, not high p-value.
The p-value only tells you how likely the observed correlation value is under the assumption that the correlation is exactly zero. In a situation that has a correlation different from zero (which is the case in almost all situations), you can make the p-value arbitrarily small simply by collecting more data, however small the correlation is.
The R function cor.test also provides a confidence interval for the correlation coefficient:
> cor.test(faithful$waiting, faithful$eruptions)
[...]
95 percent confidence interval:
 0.8756964 0.9210652

This is more informative than merely a p-value because it includes the information about the actual value of the correlation.
