Well, as Mikhail Bondarko pointed out, there is no fully self contained theory of the motives. But let me add a few things.
Generally one considers a cohomology functor $h$ from some category of schemes (like smooth, projective…) $\mathfrak{C}$ to the category of $\mathbb{Z}$ modules.
$$h: \mathfrak{C} \rightarrow \mathbb{Z}\text-\mathrm{Mod}.$$
Grothendieck's idea is then to introduce a motivic category $\mathcal{M}$, such that we have a motive functor $M:\mathfrak{C} \rightarrow \mathcal{M}$ and
a realization $r:\mathcal{M} \rightarrow \mathbb{Z}\text-\mathrm{Mod}$, with
$h = r \circ M$ i.e. the cohomology functor factors through $\mathcal{M}$.
The ingenious observation is now that in $\mathcal{M}$ the object $M(X)$ might decompose into summands of smaller motives.
The way $M(X)$ decomposes is an additional information about $X$, which can not be seen when just looking at $h(X)$.
If $h$ is the Chow functor (with integer or $\mathbb{F}_p$ coefficients)
and we are over a field $k$ of characteristic $0$, we consider the category of Chow motives $\mathcal{M}_k$ and if $X$ is a projective, homogeneous $G$-variety, for $G$ being an algebraic group over $k$, then the motivic decomposition still depends
on many invariants of $G$ (or more precise a $G$-torsor). In fact, the motivic decomposition is an invariant itself.
Let's take $G$ to be a Spin group and $X = G/P_1$, for $P_1$ the first maximal parabolic subgroup of $G$. Then $X$ is known to be a smooth, projective quadric.
Its motive is totally decomposable if and only if $G$ is split.
Otherwise there's an incredible amount of possible motivic decompositions of $X$, depending on its dimension etc.
You may now realize the issue with your questions.
What cohomology theory we are considering?
What motivic category are we working in?
Is $k$ algebraically closed (in this case Chow motives of algebraic groups are known and not really interesting for example)?
