Two-color air plasma based broadband terahertz (THz) generation and subsequent air biased coherent detection (ABCD) of the THz field are presented. Both source and detection system are characterized experimentally and numerically, yielding excellent agreement of measured and simulated signals. We reveal that it is crucial to model the whole optical setup and include the various pump distortions, like temporal and spatial walk-off as well as ellipticity of polarization, in order to interpret the experimental measurements correctly. Moreover, it turns out that geometrical effects in the ABCD scheme shape the recorded THz spectra and need to be taken into account. We confirm that THz electric fields with peak amplitude in the MV/cm range and large spectral bandwidth are produced, allowing us to demonstrate THz spectroscopy with molecular samples.