I have a time series of integer values $x \geqslant 0$. I would like to model it using, say, ARIMA, or Holt-Winters. How do I properly preprocess it for the task? I tried log-transform of $x' = x + 1$. Are there any other ways?
Use a model for non-integer time series, not ARIMA or Holt-Winters' method. You could try an INAR for example, or an autoregressive conditional Poisson model, or a GLARMA model. See Fokianos (2012) for a comprehensive review. Some possibilities in R are described in Liboschik et al (2017).