The first results from a Tully-Fisher (TF) survey of cluster galaxies are presented. The galaxies are drawn from 15 Abell clusters that lie in the redshift range similar to 9000-12,000 km s(-1) and are distributed uniformly around the celestial sky. The data set consists of R-band CCD photometry and long-slit H alpha spectroscopy. The rotation curves (RCs) are characterized by two parameters, a turnover radius r(t) and an asymptotic velocity v(a), while the surface brightness profiles are characterized in terms of an effective exponential surface brightness I-e and scale length r(e). The TF scatter is minimized when the rotation velocity is measured at r = (2.0 +/- 0.2)r(e); se significantly larger scatter results when the rotation velocity is evaluated at greater than or similar to 3 or less than or similar to 1.5 scale lengths. This effect demonstrates that RCs do not have a universal form, specified only by luminosity, as suggested by Persic, Salucci, & Stel. The scatter minimum at r = 2r(e) is interpreted in terms of a thin stellar disk plus spherical dark halo mass model. Variations in halo mass and size at fixed disk mass and size can produce extra TF scatter at arbitrary radii, but no additional scatter at the special radius r = 2r(e) provided rho(h) proportional to M-h(-0.7). In contrast to previous studies, a modest but statistically significant surface brightness dependence of the TF relation is found, v proportional to (LIe0.14)-I-0.28. This indicates a stronger parallel between the TF relation and the corresponding fundamental plane relations of elliptical galaxies than has previously been recognized. The scatter of the optimized TF relation decreases with increasing luminosity and surface brightness (SB), from similar to 0.75 mag for low-luminosity, low-SE objects to less than or similar to 0.35 mag for high-luminosity, high-SE objects. This effect is well described by a model in which the intrinsic TF scatter is similar to 0.30 mag and most of the remaining scatter is caused by rotation velocity measurement errors of similar to 15 km s(-1) independent of rotation amplitude. Future papers in this series will consider the implications of this cluster sample for deviations from uniform Hubble how on greater than or similar to 100 h(-1) Mpc scales.
