If I'm not missing something, every algebraic group is an extension of a reductive group by a unipotent group (since a reductive group is one whose unipotent radical, i.e. maximal normal unipotent connected subgroup, is trivial). Every unipotent group is rational, since it is isomorphic as a variety to $\mathbb{A}^n$. Every reductive group is rational by the Bruhat decomposition. Every Zariski-locally trivial bundle over a rational variety whose fibers are rational, is rational.

Disclaimer: I am aware of the phenomenon where I am often missing something.

The following answer applies to affine algebraic groups over algebraically closed fields, or more generally for quasi-split groups, as Scott Carnahan suggests. I don't really know about these rationality (in the number-theoretic sense) questions.

Under these conditions, every algebraic group is an extension of a reductive group by a unipotent group (since a reductive group is one whose unipotent radical, i.e. maximal normal unipotent connected subgroup, is trivial). Every unipotent group is rational, since it is isomorphic as a variety to $\mathbb{A}^n$. Every reductive group is rational by the Bruhat decomposition. Every Zariski-locally trivial bundle over a rational variety whose fibers are rational, is rational.