Here's an exercise in universal coefficients and the classification of f.g. abelian groups:
If $X$ and $Y$ are cell complexes, finite in each degree, and two maps $f_0$ and $f_1\colon X\to Y$ induce the same map on cohomology with coefficients in $\mathbb{Q}$ and in $\mathbb{F}_p$ for all primes $p$, then they induce the same map on cohomology with $\mathbb{Z}$ coefficients. Moreover, we may restrict attention to those primes $p$ such that $H^{\ast}(Y;\mathbb{Z})$ has $p$-torsion.

Applying this to $Y=BG$, for $G$ a compact Lie group, you can see that  there's no more information in the integral characteristic classes of vector bundles with structure group $G$ than there is in the rational and mod $p$ characteristic classes for those $p$ such that $H^{\ast}(BG)$ has $p$-torsion. This is advantageous because, as Jason DeVito has remarked, the structure of $H^*(BG;k)$ is simpler when $k$ is a field than when $k=\mathbb{Z}$.

Well, $H^{\ast}(BU(n))$ is torsion-free (and Chern classes generate the integer cohomology). In $H^{\ast}(BO(n))$ and $H^{\ast}(BSO(n))$ there's only 2-torsion, and Pontryagin, S-W and, in the latter case, Euler classes are all you need.  In general, there will be a finite list of special primes for each $G$.

It's sometimes worthwhile to consider the integral Stiefel Whitney classes $W_{i+1}=\beta_2(w_i)\in H^{i+1}(X;\mathbb{Z})$, the Bockstein images of the usual ones. These classes are 2-torsion, and measure the obstruction to lifting $w_i$ to an integer class. For instance, an oriented vector bundle has a $\mathrm{Spin}^c$-structure iff $W_3=0$.

[I'm sceptical of your example in $2\mathbb{CP}^2$. So far as I can see, $3a+3b$ squares to 18, not 6, and indeed, $p_1$ is not a square.]