There are many timeseries datasets that are available for practice and experimentation. One of the most famous ones is the Air Passengers dataset, containing monthly number of air passengers from 1949 to 1960. Due to the nature of it, the data has cycles and trends as shown below.
Since the variance increases over time, we're going to almost always apply the log transformation as it reduces variance. When using the data for the demonstration of an algorithm, we may or may not apply the differentiation. For example, ARIMA won't need the differentiation as differentation is part of the algorithm. See more on these techniques at stationary timeseries.
Below is the log transformed air passengers data.
For the evaluation of model performance, we're going to split the data into train and test dataset. For example, chart in the below shows an example where 80% of the data is used for training while the rest of 20% is used for testing.