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We present X-SHOOTER near-infrared spectroscopy of a large sample of 38 luminous ($M_{1450}=-29.0$ to $-24.4$) quasars at $5.78<z<7.54$, which have complementary CII observations from ALMA. This X-SHOOTER/ALMA sample provides us with the most comprehensive view of reionization-era quasars to date, allowing us to connect the quasar properties with those of its host galaxy. In this work we introduce the sample, discuss data reduction and spectral fitting, and present an analysis of the broad emission line properties. The measured FeII/MgII flux ratio suggests that the broad line regions of all quasars in the sample are already enriched in iron. We also find the MgII line to be on average blueshifted with respect to the CII redshift with a median of $-391\,\rm{km}\,\rm{s}^{-1}$. A significant correlation between the MgII-CII and CIV-CII velocity shifts indicates a common physical origin. Furthermore, we frequently detect large CIV-MgII emission line velocity blueshifts in our sample with a median value of $-1848\,\rm{km}\,\rm{s}^{-1}$. While we find all other broad emission line properties not to be evolving with redshift, the median CIV-MgII blueshift is much larger than found in low-redshift, luminosity-matched quasars ($-800\,\rm{km}\,\rm{s}^{-1}$). Dividing our sample into two redshift bins, we confirm an increase of the average CIV-MgII blueshift with increasing redshift. Future observations of the rest-frame optical spectrum with the James Webb Space Telescope will be instrumental in further constraining the possible evolution of quasar properties in the epoch of reionization.