Using a past time series to predict how a present time series will pan out? Let's say that I have past data indicating how some time series panned out. Now I also have the beginnings of a new time series that I expect to pan out in a similar trend to the old one.

What are some general data science / machine learning / statistics techniques I can use to predict the rest of the new time series? What about using some sort of online model to iteratively predict the old model's trend, and then somehow applying that to the new model?
Here you go: https://otexts.com/fpp2

But my experience is you should start using GBM based models for time series forecasting to get really good forecasts.

Check out XGBoost, Prophet, BSTS models for time series forecasting.