  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)
