@Shizuo: For $sl_n$, the situation is more explicit. The flag variety here is the set of flags $0 = V_0 \subset V_1 \cdots V_{n-1} \subset V_n = \mathbb{C}^n$ with $\mathrm{dim} V_i = i$. So the flag variety is a closed subvariety of the product of Grassmannians $Gr(1,n) \times \cdots Gr(n,n)$. Each of these Grassmannians have a explicitly Plucker embedding into the projectivization of the exterior power of $\mathbb{C}^n$. In particular, the homogeneous ideal is explicitly given by the Plucker relations. So the multi-homogeneous coordinate ring of $Gr(1,n) \times \cdots Gr(n,n)$ is just the tensor product of the known homogeneous coordinate rings.
Finally, to get the multi-homogeneous coordinate ring for the flag variety, we need to specify an incidence locus inside $Gr(1,n) \times \cdots Gr(n,n)$. Namely, we need to specify those tuples of subspaces that form a flag. But this is easy: it corresponds to certain wedge products being zero. Just impose those additional relations, i.e. mod out by the corresponding multi-homogenous ideal. Now you should have an explicit, albeit fairly long description of the homogeneous coordinate ring. The above answer for $SL_3$ looks like a special case of this construction.
$G/B$ is most naturally a multi-projective variety, embedding in the product of projectivizations of fundamental representations: $\prod \mathbb{P} (R_{\omega_i})$. So there is a multi-homogeneous coordinated ring on $G/B$. You mentioned that this ring is $\bigoplus_{\lambda\in P_+}$ $R_\lambda$. This is correct, and the grading is also apparent. It's given by the weight lattice. (To be more canonical, you should take the duals of every highest weight representation, but what you've written down is isomorphic to that.)
So all that remains is giving the multiplication law on $\bigoplus_{\lambda\in P_+}$ $R_\lambda$. You need to specify maps $R_\lambda \otimes R_\mu \mapsto R_{\lambda+\mu}$. There's a natural candidate: If you decompose $R_\lambda \otimes R_\mu$ into a direct sum of irreducible representations, $R_{\lambda+\mu}$ will appear exactly once. The multiplication law is simply projection onto this factor.