The overlap between the spectroscopic Galactic Archaeology with HERMES (GALAH) survey and Gaia provides a high-dimensional chemodynamical space of unprecedented size. We present a first analysis of a subset of this overlap, of 7066 dwarf, turn-off, and sub-giant stars. These stars have spectra from the GALAH survey and high parallax precision from the Gaia DR1 Tycho-Gaia Astrometric Solution. We investigate correlations between chemical compositions, ages, and kinematics for this sample. Stellar parameters and elemental abundances are derived from the GALAH spectra with the spectral synthesis code SPECTROSCOPY MADE EASY. We determine kinematics and dynamics, including action angles, from the Gaia astrometry and GALAH radial velocities. Stellar masses and ages are determined with Bayesian isochrone matching, using our derived stellar parameters and absolute magnitudes. We report measurements of Li, C, O, Na, Mg, Al, Si, K, Ca, Sc, Ti, V, Cr, Mn, Co, Ni, Cu, Zn, Y, as well as Ba and we note that we have employed non-LTE calculations for Li, O, Al, and Fe. We show that the use of astrometric and photometric data improves the accuracy of the derived spectroscopic parameters, especially log g. Focusing our investigation on the correlations between stellar age, iron abundance [Fe/H], and mean alpha-enhancement [α/Fe] of the magnitude-selected sample, we recover the result that stars of the high-α sequence are typically older than stars in the low-α sequence, the latter spanning iron abundances of −0.7 < [Fe/H] < +0.5. While these two sequences become indistinguishable in [α/Fe] vs. [Fe/H] at the metal-rich regime, we find that age can be used to separate stars from the extended high-α and the low-α sequence even in this regime. When dissecting the sample by stellar age, we find that the old stars (>8 Gyr) have lower angular momenta L_z than the Sun, which implies that they are on eccentric orbits and originate from the inner disc. Contrary to some previous smaller scale studies we find a continuous evolution in the high-α-sequence up to super-solar [Fe/H] rather than a gap, which has been interpreted as a separate “high-α metal-rich” population. Stars in our sample that are younger than 10 Gyr, are mainly found on the low α-sequence and show a gradient in L_z from low [Fe/H] (L_z > L_{z, ⊙}) towards higher [Fe/H] (L_z < L_{z, ⊙}), which implies that the stars at the ends of this sequence are likely not originating from the close solar vicinity.