  Absolute transition frequencies of the $2s\,^2{\rm{S}}_{1/2}$ $\rightarrow$
$2p\,^2{\rm{P}}_{1/2,3/2}$ transitions in Be$^+$ were measured with a frequency
comb in stable and short-lived isotopes at ISOLDE (CERN) using collinear laser
spectroscopy. Quasi-simultaneous measurements in copropagating and
counterpropagating geometry were performed to become independent from
acceleration voltage determinations for Doppler-shift corrections of the fast
ion beam. Isotope shifts and fine structure splittings were obtained from the
absolute transition frequencies with accuracies better than 1\,MHz and led to a
precise determination of the nuclear charge radii of $^{7,10-12}$Be relative to
the stable isotope $^9$Be. Moreover, an accurate determination of the $2p$ fine
structure splitting allowed a test of high-precision bound-state QED
calculations in the three-electron system. Here, we describe the laser
spectroscopic method in detail, including several tests that were carried out
to determine or estimate systematic uncertainties. Final values from two
experimental runs at ISOLDE are presented and the results are discussed.
