I am new to time series forecasting and have been slowly working my way through the different approaches available. I've so far mainly been using ets and arima models available in the R forecast package. As I need to forecast at a daily level taking into consideration weekends and public holidays, I have also been exploring arimax models with xreg values. As far as I can tell, it does not seem possible to use xreg values with ets models. With my sample data ets also can't forecast the step function that occurs at weekends accurately - the smoothing means the only saturday is forecast correctly but sunday is also incorrect. I cam across a series of slides from Nate Derby - http://www.sas.com/offices/NA/canada/downloads/presentations/Victoria2008/Time.pdf The final slides elude to the fact that state space models might make it easier to model timeseries. How powerful and implementable are state space models? Is this something I need to seriously consider?
To become better at time-series forecasting, it is no doubt beneficial to expand the number of forecasting methods or models available at your fingertips. Being able to model time-series data using ARIMA and exponential smoothing models is a good notch to have under your belt. Delving into non-linear models, regime switching models, and varying parameter models can only be a good thing for you.
The important thing to keep in mind is that we'd normally like to build simple linear models and not necessarily complicate matters by building non-linear models. This is one thing that you should definitely consider. That is, before considering building non-linear models, first test to see if non-linearity is actually present in the data under consideration.
You mention that you need to take into account weekends and public holidays. This suggests that maybe you need to make sure that the model you choose contains the appropriate seasonal (and non-seasonal, of course) components. Have you tried a seasonal non-seasonal multiplicate arima model? Or, a seasonal non-seasonal additive arima model?
I can only speak in general terms since you haven't given much details about the data you're working with, so a general word of advice would be to let the data do the talking and rather than choose a forecasting model a priori, let the data lead you to the appropriate model. If the data suggests that a non-linear model is worth considering, for sure, consider it.
In addition to F. Tusell's suggestion, another R package worth checking out is tsDyn. It is available at http://cran.r-project.org/web/packages/tsDyn/index.html.
Starting with the last question, yes, I think you have every reason to look at state-space models methods to solve your problem. You do not tell which software you are using, but many of them will allow you to introduce regressors, with fixed or time-varying coefficients. In R, for instance, you could look at package dlm among many others.