  The lowest-order scaling consideration of the magnetic state formation in the
Kondo lattices is performed within the $s-f$ model with inclusion of anisotropy
for both the $f-f$ coupling and $s-f$ exchange interaction. The Kondo
renormalizations of the effective transverse and longitudinal $s-f$ coupling
parameters, spin-wave frequency, gap in the magnon spectrum and ordered moment
are calculated in the case of both ferro- and antiferromagnets. The
anisotropy-driven change of the scaling behavior (e.g., critical value of $g$
for entering the strong-coupling region and the corresponding critical
exponents) is investigated numerically for N=2 and analytically in the large-N
limit. The dependence of the effective Kondo temperature on the bare $s-f$
coupling parameter $g$ weakens in the presence of anisotropy. The relative
anisotropy parameters for both the $s-f$ and $f-f$ coupling are demonstrated to
decrease during the renormalization process. The role of next-nearest exchange
interactions for this effect in the antiferromagnet is discussed.
