You're on the right track in trying to identify whether the univariate time series is stationary. There are four assumptions that typically underlie all measurement processes; namely, that the data from the process at hand "behave like":
1. Random drawings;
2. From a fixed distribution;
3. With the distribution having fixed location; and
4. With the distribution having fixed variation.
Stationary data, simply stated, is data whose mean and variance are constant over time and do not follow any trend (#3 and #4 above).
To check if these assumptions hold, you can use the following graphs:
1. Random drawings: lag plot
2. From a fixed distribution: histogram and/or normal probability plot
3. With the distribution having fixed location: run sequence plot
4. With the distribution having fixed variation: run sequence plot
In R, the stats package will provide you with functions to create these plots.
Normal Probability Plot
*qqline(x) will add a line to fit the data and will make it easier to judge whether the data follows a fixed distribution