Wherever it was possible, the effects used in 'Saving Private Ryan' were physical. Even in modern films with the best digital graphics in the world, it's usually possible to tell the difference between a CGI explosion and a real one. We're just not quite there yet with digital animation. In 1998, the differences would have been even more obvious, and so almost everything you see on screen is real.
Even with Spielberg's commitment to hyper-realism, it wouldn't be possible to actually shoot the cast, and so a little digital trickery had to be employed to provide realistic bullet holes! All such effects were added on in the edit, using a process known as Industrial Light and Magic. The effects may seem simple by modern standards, but to employ such techniques in the late 90s without them obviously seeming fake was quite an achievement.