I understand ARMA is a linear combination of lagged data points and lagged errors, but I am unclear on its implementation once parameters have been identified. Now suppose I have an ARMA model and some data. Where do I get these error terms, especially for the first term.
They are unobservable, just like errors in a regression model. You can estimate them, again just like in a regression model. Estimation of ARMA models is done via maximum likelihood, frequently via state-space representation and Kalman filtering. A good description of ARMA estimation is available Hamilton "Time Series Analysis" and other time series textbooks (and probably some threads on Cross Validated; search for