This is the first paper of a series where we study the clustering of LRG galaxies in the latest spectroscopic SDSS data release, DR6, which has 75000 LRG galaxies covering over 1 $Gpc^3/h^3$ at $0.15<z<0.47$. Here we focus on modeling redshift space distortions in $xips$, the 2-point correlation in separate line-of-sight and perpendicular directions, on large scales. % and away from the line-of-sight. 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% larger than the Poisson error commonly used. We first use the normalized quadrupole for the whole sample (mean z=0.34) to estimate $beta=f(Omega_m)/b=0.34 pm 0.03$, where $f(Omega_m)$ 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 $xips$ plane to find $Omega_{0m}= 0.245 pm 0.020$ (h=0.72) and the biased amplitude $b sigma_8 = 1.56 pm 0.09$. For standard gravity, we can combine these measurements to break degeneracies and find $sigma_8=0.85 pm 0.06$, $b=1.85 pm 0.25$ and $f(Omega_m)=0.64 pm 0.09$. We present constraints for modified theories of gravity and find that standard gravity is consistent with data as long as $0.80<sigma_8<0.92$. 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. (Abridged)
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