We report BV(RI)C data for a select group of 14 Cepheids in the Large Magellanic Cloud and eight in the Small Magellanic Cloud that have preexisting radial velocity curves. The photometry was obtained as part of a program to determine distances to these Cepheids by means of the visual surface brightness technique and to improve significantly the optical BV(RI)C light curves of Magellanic Cloud Cepheids. The data were acquired on the 0.9 m telescope at Cerro Tololo Inter-American Observatory with the CFCCD instrument and with the 1 m photometric telescope at ESO using aperture photometry. The median number of measures per Cepheid is 46 in each band, and the uncertainty in the photometry is typically 0.01 mag. Using these data together with data from the literature, we determined improved periods for all variables. For most of the Cepheids, these revised periods lead to excellent, low-noise light curves, but for a few variables, the periods are obviously variable. Introduction: In this paper we present the new photometric data, demonstrate the consistency among the data obtained from CCD and aperture photometry, and determine improved periods for all of the variables. In follow-up papers we will use these observations to determine individual Cepheid distances and radii from the visual surface brightness technique and also from the infrared surface brightness technique (Welch 1994AJ....108.1421W; Fouque & Gieren 1997A&A...320..799F). Table 1 gives our final photometric results. All entries in Table 1 having B magnitudes are CCD observations (CTIO), and the ones lacking B magnitudes are photoelectric observations (ESO). The uncertainties in the CCD results are dominated by the uncertainty in the aperture correction. The standard deviation in the comparison stars on a particular image is typically +/-0.004 mag and always less than +/-0.01 mag. The aperture correction to the zero point of each frame is estimated to be better than +/-0.01 mag. We adopt +/-0.01 mag as the typical uncertainty in all magnitudes. For the photoelectric observations, we inferred the uncertainties from scatter in the observations of the Cepheids themselves, finding +/-0.010 mag in V and +/-0.012 mag in R and I. We combined our new photometry with existing photoelectric photometry from the literature to determine a new period for each Cepheid. This was greatly aided by the McMaster Cepheid Data Archive maintained by Doug Welch at http://www.physics.mcmaster.ca/Cepheid/. (The data reported here will be added to the McMaster archive.) Periods were found using an algorithm due to T. Deeming given in Bopp et al. (1970MNRAS.147..355B) and are listed in Table 2. The uncertainty of each period is reflected in the number of digits given in the table. For most of the Cepheids in our sample, the new periods lead to excellent light curves of very low scatter, which indicates that the periods for these variables are both correct and stable over the time of our observations. For the Cepheids with comments in Table 2, however, the periods appear to be changing in time. The existing data set is insufficient to describe fully the nature of these period changes. Also in Table 2 we compare our intensity mean V magnitudes with those from the literature, as represented by an extensive, private data base maintained by J.A.R. Caldwell (1997, private communication). The intensity means were determined from Fourier fits to the (intensity-converted) light curves. The agreement is very good. The mean difference between the data sets is -0.005 mag, and the standard deviation is 0.019 mag. For Cepheids with the best new data, we have listed a value to 0.001 mag, while for stars with less complete light curves, the values are listed to 0.01 mag only. Table 2: New Periods and Mean V Magnitudes for Magellanic Cloud Cepheids Period V(new) V(SAAO) Cepheid (days) Comment (mag) (mag) --------------------------------------------------------- LMC --------------------------------------------------------- HV 879... 36.827 13.332 13.35 HV 883... 133.585 12.14 HV 899... 31.0461 13.433 13.43 HV 900... 47.5085 12.766 12.78 HV 909... 37.5654 12.774 12.74 HV 2257... 39.3699 13.031 13.06 HV 2338... 42.1944 12.759 12.78 HV 2447... 118.3115 11.990 12.00 HV 2827... 78.77 Variable? 12.280 12.30 HV 2864... 10.9845 14.651 14.65 HV 2883... 108.968 12.41 HV 5497... 98.975 Variable? 11.92 11.94 HV 12815.. 26.1157 13.506 13.48 HV 12816.. 9.1089 14.500 14.52 --------------------------------------------------------- SMC --------------------------------------------------------- HV 821... 127.314 Variable? 11.93 11.94 HV 824... 65.8635 12.36 12.37 HV 829... 85.199 Variable? 11.93 11.91 HV 834... 73.639 Variable? 12.21 12.21 HV 837... 42.6954 13.24 13.23 HV 1338... 8.49503 15.12 15.16 HV 1365... 12.4117 Variable? 15.04 15.02 HV 11157.. 69.0872 12.93 12.94