Observations in different spectral lines give us information about the different layers of the solar atmosphere. Here we analyze the propagation of waves in sunspots from the photosphere to the chromosphere using time series of cospatial Ca II H intensity spectra and polarimetric spectra of Si I 10827 A and He I 10830 A multiplet. From the Doppler shifts of these lines we retrieve the temporal variations of the velocity along the line-of-sight at several heights. Phase spectra are used to get the relation between oscillatory signals measured at each spectral signature. Our analysis reveals standing waves for frequencies lower than 3.5 mHz and propagating waves for higher frequencies, which steepen into shocks in the chromosphere. Oscillations are detectable in Ca II H wings and they are propagated along line wing layers to the line core. Ca II H core forms at a lower height than the He I 10830 A line. A time delay of about 30 s is measured between the Doppler signals detected at both wavelengths. We also find that in "cold" sunspots the Si I 10827 A forms deeper than in the quiet sun. This type of measurements demonstrate the importance of simultaneous co-spatial observations at different wavelengths. Future infrastructures, such as GREGOR and EST, should include multi-wavelength capabilities to make possible the study of the photosphere-chromosphere connection with the highest spatial and temporal resolution.