Which number is of value in linear regression: R Squared or P? I am attempting to compare a two pitching stats (W/L% and ERA) to determine to what extent the latter can predict the former. After entering my data and performing the appropriate linear regression analysis I am left with two seemingly contradictory results.
1) I have an R-sq value of .257 which would lead me to state that the relationship is fairly weak
2) I have a P-value indistinguishable from 0 which would lead me to state that ERA is a near perfect predictor of W/L%
What do I not understand about the meaning of these two numbers? What do they tell me about the correlation?
 A: Point 2) is incorrect in your first paragraph. A p-value does not tell you that one variable is a good, bad or near perfect predictor of another. It tells you precisely this:

If, in the population from which this sample was drawn, there really
  was no effect, how likely is it that we would get a test statistic
  this large or larger in a sample of this size?

It also may not true that a linear regression is appropriate. WL % is bounded (0 to 100). It may be better to use a regression method (such as beta regression) that takes account of this. Also, if you have data over more than one season, you need to account for that and if you have data from multiple players on one team, you will want to account for that. 
However, the short answer to the question in your title is "Both". 
A: 
Which number is of value in linear regression: R Squared or P?

Depending on circumstances, either, both, or neither may be valuable, informative or interesting.   

I am attempting to compare a two pitching stats (W/L% and ERA) to determine to what extent the latter can predict the former. After entering my data and performing the appropriate linear regression analysis I am left with two seemingly contradictory results. 
1) I have an R-sq value of .257 which would lead me to state that the relationship is fairly weak 

What counts as 'weak' correlation is somewhat context dependent. In chemistry, 0.9 might be regarded as weak, while in psychology 0.4 might be regarded as pretty reasonable (your correlation is above 0.5). What it does tell you is that most of the variation in W/L% is not due to ERA.

2) I have a P-value indistinguishable from 0 which would lead me to state that ERA is a near perfect predictor of W/L%

This is not remotely a reasonable interpretation of a very small p-value. It is possible - with large enough sample sizes - to get a close-to-zero p-value with something whose correlation is below 0.01. All the small p-value tells you is that the (possibly weak) correlation that is there is clearly distinguishable from noise or chance; something is actually going on there.

What do I not understand about the meaning of these two numbers? 

In particular you should check the definition of the p-value, right at the start of the Wikipedia article:

In statistical significance testing the p-value is the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true.

You may also want to check the meaning of R^2.

What do they tell me about the correlation?

$R^2$ is the square of the correlation. The p-value on its own tells you nothing about the correlation (though if you know the p-value and the sample size, you could back out the F-value and hence the $R^2$ from that).
--
W/L% is a proportion - a scaled count. 
As such, a linear model is highly unlikely to be suitable, and measures of linear correlation may be misleading about the strength of the association. Further, several of the other assumptions of linear regression are likely to be unsuitable (such as equality of variance).
You may be better off considering a binomial or quasi-binomial GLM.
