We present adaptive optics (AO) assisted SINFONI integral field unit (IFU) spectroscopy of 11 Hα emitting galaxies selected from the High-Z Emission Line Survey (HiZELS). We obtain spatially resolved dynamics on ∼kpc-scales of star-forming galaxies [stellar mass M⋆ = 109.5 − 10.5 M⊙ and star formation rate (SFR) = 2–30 M⊙ yr−1] near the peak of the cosmic star formation rate history. Combining these observations with our previous SINFONI-HiZELS campaign, we construct a sample of 20 homogeneously selected galaxies with IFU AO-aided observations – the ‘SHiZELS’ survey, with roughly equal number of galaxies per redshift slice, at z = 0.8, 1.47 and 2.23. We measure the dynamics and identify the major kinematic axis by modelling their velocity fields to extract rotational curves and infer their inclination-corrected rotational velocities. We explore the stellar mass Tully–Fisher relationship, finding that galaxies with higher velocity dispersions tend to deviate from this relation. Using kinemetry analyses, we find that galaxy interactions might be the dominant mechanism controlling the star formation activity at z = 2.23 but they become gradually less important down to z = 0.8. Metallicity gradients derived from the [N ii]/Hα emission line ratio show a median negative gradient for the SHiZELS survey of Δlog(O/H)/ΔR = −0.026 ± 0.008 dex kpc−1. We find that metal-rich galaxies tend to show negative gradients, whereas metal-poor galaxies tend to exhibit positive metallicity gradients. This result suggests that the accretion of pristine gas in the periphery of galaxies plays an important role in replenishing the gas in ‘typical’ star-forming galaxies.