If you want to do time series prediction and you have a sample of a univariate time series at hand, you would start by plotting it and familiarizing yourself with it. Then you would try to find patterns with the hope of extrapolating them into the future1. There could be, for example,
- seasonality -- in conditional mean or variance (or even higher-order moments);
- autoregressive conditional heteroskedasticity.
To discover these patterns, you could use
- seasonal plots (slice the time series into full periods, e.g. years, and plot those on top of each other); function
seasonplot in "forecast" package in R;
- (partial) autocorrelation plots (ACF and PACF); functions
pacf in R;
- (partial) autocorrelation plots (ACF and PACF) on squared mean-adjusted data.
Once you have identified some patterns, you could then start developing models.
1Beware of outliers and nonstationary behaviour, e.g. long-lasting changes in level or variance of the series. When neglected, they could seriously affect your diagnostics of seasonality, autocorrelation and autoregressive conditional heteroskedasticity. Outliers and nonstationarities should be dealt with first.