BaAg₂Cu[VO₄]₂ contains Cu(II) S=1/2 ions on a distorted two-dimensional triangular lattice interconnected via non-magnetic [VO₄] entities. DFT band structure calculations, quantum Monte-Carlo simulations, and high-field magnetization measurements show that the magnetism of this compound is determined by a superposition of ferromagnetic (FM) and antiferromagnetic (AFM) uniform spin-1/2 chains with nearest neighbor exchange couplings of J^FM=-19 K and J^AFM=9.5 K (A. Tsirlin, A. Möller, B. Lorenz, Y. Skourski, H. Rosner, Phys. Rev. B 85 (2012) 014401). Here we report the study of BaAg₂Cu[VO₄]₂ by high-field/frequency electron spin resonance (HF-ESR) and nuclear magnetic resonance (NMR) spectroscopies, which probe the local magnetic properties. In the HF-ESR measurements, we observe an anisotropic ESR spectrum typical for the Cu(II) ions and determine the g-tensor, g_∥=2.38 and g_⊥=2.06. Moreover, we see a substantial change in the spectral shape of the ESR lines at low temperatures indicating the presence of short range magnetic correlations. The analysis of the low-temperature ESR spectra shows that its peculiar structure is due to the development of the anisotropic internal fields corresponding to FM and AFM correlations in the respective Cu spin chains. In the NMR spectra the signals from ⁵¹V nuclei in the two types of chains were identified. The analysis of the temperature evolution of these signals strongly supports the ESR findings on the occurrence of two types of Cu chains. Altogether, the HF-ESR and NMR results confirm theoretical predictions of the superposition of FM and AFM Cu(II) spin-1/2 chains in the studied material.