For each type of t-test you do, one should always report the t-statistic, df, and p-value, regardless of whether the p-value is statistically significant (< 0.05). A succinct notation, including which type of test was done, is:
one-sample t(df) = t-value, p = p-value
two-sample t(df) = t-value, p = p-value
paired t(df) = t-value, p = p-value
where "df", "t-value", and "p-value" are replaced by their measured values. Regarding the number of digits to report, we are primarily concerned with whether p is greater than or less than 0.05; so as a rule of thumb, one need only report one digit behind the decimal for a t-value, and report two digits behind the decimal for a p-value (one could go to three if the p-value is near 0.05, such as 0.045 or 0.055).
Negative t-values: The sign of a t-value tells us the direction of the difference in sample means, which can be difficult to interpret without further explanation: Does a negative t-value indicate A's sample mean was greater than B's, or less? Therefore, it is common to report the t-value as the absolute value of the t-value given by the statistics program. If you do this, be sure to indicate the direction of the mean-difference (even if nonsignificant) in some other way, such as by mentioning the sample means in the text, or by showing the sample means graphically, as in a bar chart.
When p < 0.05, then "the magnitude of the effect" becomes of interest. So in that case, in addition to t, df, and p, one should report the sample mean or means, or the mean difference, mentioned above. If you report a mean difference, be clear about the direction of the effect: "A took, on average, 1.2 seconds less time than B", for instance.
The number of digits reported as an "effect size" shouldn't exceed the accuracy that the variable was measured at (don't imply microsecond precision, for instance, if you only measured to the nearest millisecond). And, remember that, unlike t, df, and p which are pure unitless numbers, these means will have the units of the original measurements. Do not forget to include these units.