In the age of high-resolution spectroscopic surveys of stars in the Milky Way, the number of stars for which detailed abundances of multiple elements is available is rapidly increasing. These elemental abundances are directly influenced by the evolutionary history of the Galaxy, but this can be difficult to interpret without an absolute timeline of the abundance enrichment. We present age-abundance trends for [M/H], [α/M], and 17 individual elements using a sample of 700 solar neighbourhood Hipparcos red giant stars observed by APOGEE. These age trends are determined through a Bayesian hierarchical modelling method presented by Feuillet et al. (2016). We confirm that the [α/M]-age relation in the solar neighbourhood is steep and relatively narrow (0.20 dex), as are the [O/M]- and [Mg/M]-age relations. The age trend of [C/N] is steep and smooth, consistent with stellar evolution. The [M/H]-age relation has a mean age dispersion of 0.28 dex and a complex overall structure. Within each 0.1 dex bin of [M/H], the [α/M]-age trend is similar to the global trend. The oldest stars in our sample are those with the lowest and highest metallicities, while the youngest stars are those with solar metallicity. These results provide strong constraints on theoretical models of Galactic chemical evolution (GCE). We compare them to the predictions of one-zone GCE models and multi-zone mixtures, both analytic and numerical. These comparisons support the hypothesis that the solar neighbourhood sample is composed of stars born at a range of Galactocentric radii, and that the most metal-rich stars likely migrated from a region with earlier and more rapid star formation such as the inner Galaxy.