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).

Documentation forcor.test in R says "By default, probabilities for method="spearman" and method="kendall" are found by normal theory. If normal=="FALSE", then repetitive calls are made to cor.test. ..."
What attention have you given to normality of data based on "(number of characters, number of cuts, IMDB rating etc.)"?
From what is said in your Question and in some of the Comments, I would wonder which P-values should be taken seriously which should not.
