We know from previous work that there are substantial differences between powerful FR I and FR II radio galaxies. In this paper we look at the correlations of line luminosity, radio luminosity, core radio power, and host galaxy optical magnitude independently for FR I and FR II radio galaxies and compare these correlations with those for an optically selected control sample of early-type galaxies. In this, Paper I in a two-paper series, we list the principal results; in Paper II we discuss the implications of these results for our understanding of the FR I-FR II dichotomy and the central engines of powerful radio galaxies. Our principal results are the following. Correlating core power to total power we find: (1) The difference between the radio core powers of FR I and FR II galaxies is less than the difference between the extended radio powers: the median total radio powers of FR II galaxies in our sample are about 40 times greater than those of FR I galaxies, while the median radio core powers of FR II galaxies in our sample are only about 4 times greater than those of FR I galaxies. (2) In agreement with previous work, we find a decrease (with slope -0.38 +/- 0.05) in the ratio of core to lobe radio power, the R-parameter, with increasing total radio power. However, there is a significant scatter (2 orders of magnitude) in the R-parameter. (3) At a fixed total radio power, FR I and FR II galaxies have the same ratio of core to lobe power and a comparable scatter. We investigate the possibility of systematic effects skewing these results but find no evidence for this: (1) The R-parameter is not affected by redshift selection effects. (2) Orientation and beaming effects are either dominated by the large intrinsic scatter in the R-parameter, or they themselves produce that scatter. Since the scatter is the same for FR I and FR II sources, beaming effects must be equally important or unimportant in both FR types. We analyze the three-way correlation between redshift, radio power, and emission-line luminosity and find that the correlation of radio power with redshift is stronger than that of emission-line luminosity with redshift. In fact, for the FR I galaxies there is virtually no correlation between line luminosity and redshift. When correlating the total radio power to the emission-line luminosity for our full sample of radio-loud galaxies, we find that these two parameters are strongly correlated over 8 orders of magnitude in emission-line luminosity and 10 orders of magnitude in total radio power. When we look independently at the correlations for FR I and FR II radio galaxies, we find: (1) Each FR type has its own independent correlation of radio power and emission-line luminosity. (2) The functional relationships between emission-line luminosity (L(line)) ,and radio power (P-408 (MHz)) are different; L(line) = (0.75 +/- 0.08) x P-408 (MHz) + (14.8 +/- 2.3) for FR II galaxies and L(line) = (0.28 +/- 0.07) x P-408 (MHz) + (26.3 +/- 1.8) for FR I galaxies. (3) The FR I and FR II radio sources are offset with respect to one another in the radio-line luminosity plane, which can be described in two fashions: for the same total radio power as the FR I galaxies, the FR II galaxies produce consistently about 5-30 times as much emission-line luminosity; or, for the same amount of emission-line activity as the FR II galaxies, the FR I galaxies produce about 10-100 times as much total radio power. (4) For FR I sources and optically selected sources, we find a correlation of line luminosity with host galaxy optical magnitude (there is none for FR II galaxies). Removing the dependence of line luminosity on host galaxy optical magnitude, we find a 2 sigma dependence of line luminosity on radio luminosity for FR I sources. Correlating the core radio powers to the emission-line luminosities, we find: (1) Each FR type has its own correlation between the radio core power and the emission-line luminosity. Both correlations are as good as those between total power and emission-line luminosity. (2) The functional relationships for FR I and FR II sources are again different; L(line) (0.62 +/- 0.10) x P-core + (19.9 +/- 2.3) for FR II galaxies and L(line) = (0.30 +/- 0.12) x P-core + (26.3 +/- 2.8) for FR I galaxies. (3) The two FR types separate out even more clearly in the core radio power-emission-line luminosity plane than in the total radio power-line luminosity plane. For the same radio core power as the FR I galaxies, the FR II galaxies produce consistently about 10-50 times as much emission-line luminosity; or, for the same amount of emission-line activity as the FR II galaxies, the FR I galaxies produce about 200-300 times as much radio core power. The most important result is that FR I and FR II radio sources display strong differences in their correlations of line luminosity, ratio total and core power, and host galaxy optical magnitude. These differences may reflect fundamental differences in the properties of the central engines in these two types of radio galaxies. A detailed discussion of the implications is deferred to Paper II.
