This paper reports on trapping and laser cooling of singly ionized strontium ions in a linear Paul trap. We describe the loading technique based on two-photon absorption of femtosecond pulses at a wavelength of 431 nm and we compare this technique to electron-bombardment ionization. We perform Doppler cooling of the ions in the few-ions regime as well as in the case of large clouds. The analysis of the fluorescence spectra and direct imaging of the cloud allows us to identify the Coulomb-crystal regime. These experiments open the way to the use of a large cold trapped-ion cloud as a medium for quantum optics and quantum information experiments.