We present a spectroscopic survey of 230 white dwarf candidates within 40pc of the Sun from the William Herschel Telescope and Gran Telescopio Canarias. All candidates were selected from Gaia Data Release 2 (DR2) and in almost all cases, had no prior spectroscopic classifications. We find a total of 191 confirmed white dwarfs and 39 main-sequence star contaminants. The majority of stellar remnants in the sample are relatively cool (<Teff>=6200K), showing either hydrogen Balmer lines or a featureless spectrum, corresponding to 89 DA and 76 DC white dwarfs, respectively. We also recover two DBA white dwarfs and 9-10 magnetic remnants. We find two carbon-bearing DQ stars and 14 new metal-rich white dwarfs. This includes the possible detection of the first ultra-cool white dwarf with metal lines. We describe three DZ stars for which we find at least four different metal species, including one that is strongly Fe- and Ni-rich, indicative of the accretion of a planetesimal with core-Earth composition. We find one extremely massive (1.31+/-0.01M_{sun}_) DA white dwarf showing weak Balmer lines, possibly indicating stellar magnetism. Another white dwarf shows strong Balmer line emission but no infrared excess, suggesting a low-mass sub-stellar companion. A high spectroscopic completeness (>99 per cent) has now been reached for Gaia DR2 sources within 40-pc sample, in the Northern hemisphere ({delta}>0{deg} and located on the white dwarf cooling track in the Hertzsprung-Russell diagram. A statistical study of the full northern sample is presented in a companion paper (McCleery et al., Paper II, 2020MNRAS.499.1890M, Cat. J/MNRAS/499/1890).