There are generally two ways to state statistical results succinctly. In both cases, a statement is made and the statistical results are added parenthetically as support. The statistical results typically include the statistic, the degrees of freedom, and the p-value, in that order. Note that you should report only a few significant figures, generally fewer than what R reports. Not only are those extra figures likely not relevant and inaccurate, they also make your results harder to read.
The first style puts the emphasis on the null hypothesis, making it is somewhat indirect. For example, “The null hypothesis that the means of the two groups are equal was accepted (t=-0.37, df=29.4, p=0.72)”. Most people do not use this style, and prefer the second style, which I also recommend for its clarity.
The second style puts the emphasis on the results, without being so formal about the null hypothesis. For example, “The means of the two groups are not statistically different (t=-0.37, df=29.4, p=0.72)”. Note that this might also be worded “The means of the two groups are not significantly different (t=-0.37, df=29.4, p=0.72)” or “The means of the two groups are statistically indistinguishable (t=-0.37, df=29.4, p=0.72)”.
Confidence intervals are stated similarly, giving the estimate and the confidence interval. For example, “Sunlight and chlorophyll have a positive non-zero correlation (r=0.64, 95% C.I.: 0.29–0.85)”. This might also be stated “Sunlight and chlorophyll are significantly correlated (r=0.64, 95% C.I.: 0.29–0.85)” or less directly, “The correlation between sunlight and chlorophyll is statistically significant (r=0.64, 95% C.I.: 0.29–0.85)”.
Note that, regardless of whether you opt for p-values or for confidence intervals, you should be able to state the conclusion succinctly, supported by the necessary statistics, which are provided parenthetically with an appropriate number of significant figures.