We present the first measurements of Lyman-α (Lyα) forest correlations using early data from the Dark Energy Spectroscopic Instrument (DESI). We measure the auto-correlation of Lyα absorption using 88 509 quasars at z > 2, and its cross-correlation with quasars using a further 147 899 tracer quasars at z ≳ 1.77. Then, we fit these correlations using a 13-parameter model based on linear perturbation theory and find that it provides a good description of the data across a broad range of scales. We detect the BAO peak with a signal-to-noise ratio of 3.8σ, and show that our measurements of the auto- and cross-correlations are fully-consistent with previous measurements by the Extended Baryon Oscillation Spectroscopic Survey (eBOSS). Even though we only use here a small fraction of the final DESI dataset, our uncertainties are only a factor of 1.7 larger than those from the final eBOSS measurement. We validate the existing analysis methods of Lyα correlations in preparation for making a robust measurement of the BAO scale with the first year of DESI data.