  (abridged) DYNAMO is a multi-wavelength, spatially-resolved survey of local
($z \sim 0.1$) star-forming galaxies designed to study evolution through
comparison with samples at z~2. Half of the sample has integrated H-alpha
luminosities of >$10^{42}$ erg/s, the typical lower limit for resolved
spectroscopy at z~2. The sample covers a range in stellar mass
($10^9$-$10^{11}$ Msun) and star-formation rate (0.2-100 Msun/yr). In this
first paper of a series, we present integral-field spectroscopy of H-alpha
emission for the sample of 67 galaxies. We infer gas fractions in our sample as
high as ~0.8, higher than typical for local galaxies. Gas fraction correlates
with stellar mass in galaxies with star-formation rates below 10 Msun/yr, as
found by COLDGASS, but galaxies with higher star-formation rates have higher
than expected gas fractions. There is only a weak correlation, if any, between
gas fraction and gas velocity dispersion. Galaxies in the sample visually
classified as disc-like are offset from the local stellar-mass Tully-Fisher
relation to higher circular velocities, but this offset vanishes when both gas
and stars are included in the baryonic Tully-Fisher relation. The mean gas
velocity dispersion of the sample is ~50 km/s, and V/sigma ranges from 2 to 10
for most of the discs, similar to 'turbulent' galaxies at high redshift. Half
of our sample show disc-like rotation, while ~20 percent show no signs of
rotation. The division between rotating and non-rotating is approximately...
