More than 100 millisecond pulsars (MSPs) have been discovered in radio observations of gamma-ray sources detected by the Fermi Large Area Telescope (LAT), but hundreds of pulsar-like sources remain unidentified. Here, we present the first results from the targeted survey of Fermi-LAT sources being performed by the Transients and Pulsars with MeerKAT (TRAPUM) Large Survey Project. We observed 79 sources identified as possible gamma-ray pulsar candidates by a Random Forest classification of unassociated sources from the 4FGL catalogue. Each source was observed for 10 min on two separate epochs using MeerKAT's L-band receiver (856-1712 MHz), with typical pulsed flux density sensitivities of $\sim 100\, \mu$Jy. Nine new MSPs were discovered, eight of which are in binary systems, including two eclipsing redbacks and one system, PSR J1526-2744, that appears to have a white dwarf companion in an unusually compact 5 h orbit. We obtained phase-connected timing solutions for two of these MSPs, enabling the detection of gamma-ray pulsations in the Fermi-LAT data. A follow-up search for continuous gravitational waves from PSR J1526-2744 in Advanced LIGO data using the resulting Fermi-LAT timing ephemeris yielded no detection, but sets an upper limit on the neutron star ellipticity of 2.45 × 10-8. We also detected X-ray emission from the redback PSR J1803-6707 in data from the first eROSITA all-sky survey, likely due to emission from an intrabinary shock.