This is the first paper of a series where we study the clustering of LRG galaxies in the latest spectroscopic Sloan Digital Sky Survey (SDSS) data release, DR6, which has 75 000 LRG galaxies covering over 1 Gpc3 h−3 at 0.15 < z < 0.47. Here we focus on modelling redshift-space distortions in ξ(π, σ), the two-point correlation function in separate line of sight and perpendicular directions, on large scales. We use large mock simulations to study the validity of models and errors. We show that errors in the data are dominated by a shot-noise term that is 40 per cent larger than the Poisson error commonly used. We first use the normalized quadrupole for the whole sample (mean z= 0.34) to estimate β=f(Ωm)/b= 0.34 ± 0.03, where f(Ωm) is the linear velocity growth factor and b is the linear bias parameter that relates galaxy to matter fluctuations on large scales. We next use the full ξ(π, σ) plane to find Ω0m= 0.245 ± 0.020 (h= 0.72) and the biased amplitude bσ8= 1.56 ± 0.09. For standard gravity, we can combine these measurements to break degeneracies and find σ8= 0.85 ± 0.06, b= 1.85 ± 0.25 and f(Ωm) = 0.64 ± 0.09. We present constraints for modified theories of gravity and find that standard gravity is consistent with data as long as 0.80 < σ8 < 0.92. We also calculate the cross-correlation with WMAP5 and show how both methods to measure the growth history are complementary to constrain non-standard models of gravity. Finally, we show results for different redshift slices, including a prominent BAO peak in the monopole at different redshifts. The ξ(π, σ) data on large scales is shown to be in remarkable agreement with predictions and shows a characteristic large region of negative correlation in the line of sight, a BAO ring and a prominent radial BAO peak. The significance of this is presented in Paper IV of this series. We include a study of possible systematic effects in our analysis to find that these results are quite robust.