A researcher reports r = .60, p =.02 when correlating the relationship between anxiety level and ...

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Rationale 1: You are correct in concluding that r = .60 is a positive correlation; if the variable of anxiety increased, there was also likely to be a rise in pulse in many cases. An r of .60 is considered to be a moderate correlation and a p value of .02 is less than the preset of .05, so the difference is statistically significant.
Rationale 2: A negative correlation means that one variable increased and the other decreased. This is usually denoted by a negative (-) sign in front of the value reported.
Rationale 3: An r = .60 is positive, meaning if one variable increased the other was also likely to increase. This value also indicates a moderate correlation; so although it did not happen in all cases, the tendency for both to increase together would still be visible if you were to plot the relationship out on a scatter plot. The p value is less than , so the difference did reach significance.
Rationale 4: Correlations of .30 or less are usually considered weak, although one should remember that you can have a weak yet statistically significant correlation; the two factors are not mutually exclusive.