We present new intensity and polarization observations of the Taurus Molecular Cloud (TMC) region in the frequency range 10-20 GHz with the multifrequency instrument (MFI) mounted on the first telescope of the Q-U-I-JOint TEnerife (QUIJOTE) experiment. From the combination of the QUIJOTE data with the WMAP 9-yr data release, the Planck second data release, the DIRBE maps, and ancillary data, we detect an anomalous microwave emission (AME) component with flux density S_{AME, peak} = 43.0 ± 7.9 Jy in the TMC and S_{AME, peak} = 10.7 ± 2.7 Jy in the dark cloud nebula L1527, which is part of the TMC. In the TMC the diffuse AME emission peaks around a frequency of 19 GHz, compared with an emission peak about a frequency of 25 GHz in L1527. In the TMC, the best constraint on the level of AME polarization is obtained at the Planck channel of 28.4 GHz, with an upper limit π _AME< 4.2 {{ per cent}} (95 {{ per cent}} C.L.), which reduces to π _AME< 3.8 {{ per cent}} (95 {{ per cent}} C.L.) if the intensity of all the free-free, synchrotron and thermal dust components are negligible at this frequency. The same analysis in L1527 leads to π _AME< 5.3{{ per cent}} (95 {{ per cent}} C.L.) or π _AME< 4.5 {{ per cent}} (95 {{ per cent}} C.L.) under the same assumption. We find that in the TMC and L1527 on average about 80{{ per cent}} of the H II gas should be mixed with thermal dust. Our analysis shows how the QUIJOTE-MFI 10-20 GHz data provide key information to properly separate the synchrotron, free-free, and AME components.