The results of a Wilcoxon test only make sense when the pairs are independent – that whatever factor caused a difference (between paired values) to be too high or too low affects only that one pair. Prism cannot test this assumption. You must think about the experimental design. For example, the errors are not independent if you have six pairs of values, but these were obtained from three animals, with duplicate measurements in each animal. In this case, some factor may cause the after-before differences from one animal to be high or low. This factor would affect two of the pairs (but not the other four), so these two are not independent.
If the P value is large (say larger than 0.05), you should question whether it made sense to use a paired test. Your choice of whether to use a paired test or not should not be based solely on this one P value, but also on the experimental design and the results you have seen in other similar experiments.
Use the Wilcoxon test only to compare two groups. To compare three or more matched groups, use the Friedman test followed by post tests. It is not appropriate to perform several Wilcoxon tests, comparing two groups at a time.
If you chose a one-tail P value, you should have predicted which group would have the larger median before collecting any data. Prism does not ask you to record this prediction, but assumes that it is correct. If your prediction was wrong, then ignore the P value reported by Prism and state that P>0.50.
By selecting a nonparametric test, you have avoided assuming that the data were sampled from Gaussian distributions. But there are drawbacks to using a nonparametric test. If the populations really are Gaussian, the nonparametric tests have less power (are less likely to give you a small P value), especially with small sample sizes. Furthermore, Prism (along with most other programs) does not calculate confidence intervals when calculating nonparametric tests. If the distribution is clearly not bell-shaped, consider transforming the values (perhaps to logs or reciprocals) to create a Gaussian distribution and then using a t test.
The Wilcoxon test first computes the difference between the two values in each row, and analyzes only the list of differences. The Wilcoxon test does not assume that those differences are sampled from a Gaussian distribution. However it does assume that the differences are distributed symmetrically around their median.