Nitrogen-vacancy (NV) quantum defects in diamond are sensitive detectors of magnetic fields. Owing to their atomic size and optical readout capability, they have been used for magnetic resonance spectroscopy of nanoscale samples on diamond surfaces. Here, we present a protocol for fabricating NV diamond chips and for constructing and operating a simple, low-cost 'quantum diamond spectrometer' for performing NMR and electron spin resonance (ESR) spectroscopy in nanoscale volumes. The instrument is based on a commercially available diamond chip, into which an NV ensemble is ion-implanted at a depth of ~10 nm below the diamond surface. The spectrometer operates at low magnetic fields (~300 G) and requires standard optical and microwave (MW) components for NV spin preparation, manipulation, and readout. We demonstrate the utility of this device for nanoscale proton and fluorine NMR spectroscopy, as well as for the detection of transition metals via relaxometry. We estimate that the full protocol requires 2-3 months to implement, depending on the availability of equipment, diamond substrates, and user experience.