We present the X-ray properties of a sample of 17 radio sources observed with the Chandra X-ray Observatory as part of a project aimed at studying the X-ray emission from their radio jets . In this paper , we concentrate on the X-ray properties of the unresolved cores . The sample includes 16 quasars ( 11 core-dominated and 5 lobe-dominated ) in the redshift range z =0.30–1.96 , and one low-power radio-galaxy at z =0.064 . No diffuse X-ray emission is present around the cores of the quasars , except for the nearby low-power galaxy that has diffuse emission on a scale and with a luminosity consistent with other FRIs . No high-amplitude , short-term variability is detected within the relatively short Chandra exposures . However , 1510 - 089 shows low-amplitude flux changes with a timescale of \sim 25 minutes . The X-ray spectra of the quasar cores are generally well described by a single power law model with Galactic absorption . However , in six quasars we find soft X-ray excess emission below 1.6 keV . Interestingly , we detect an Fe K-shell emission line , consistent with fluorescent K \alpha emission from cold Iron , in one lobe- and two core-dominated sources . The average X-ray photon index for the quasars in the sample is \Gamma _ { sample } = 1.66 and dispersion \sigma _ { sample } = 0.23 . The average spectral slope for our sample is flatter than the slope found for radio-quiet quasars and for radio-loud AGNs with larger jet orientations ; this indicates that beaming affects the X-ray emission from the cores in our sample of quasars .