A proper analysis begins with looking at descriptive statistics. From examining the mean, standard deviations and quartiles of the data, you are able to get a rough idea of what your data looks like. Following this up with graphical techniques, such as, histograms, scatterplots or even Q-Q plots can give you an idea how your data is distributed and if you are going to run into problems later when you must meet the assumptions of other analysis techniques.
If your data set comes from an experiment, where you might have different groups of subjects that are placed into experimental cells, then ANOVA might be the ticket. This technique compares the means of different experimental groups. For example, if you were to have an experiment in which you gave a placebo or 10, 20, or 30 milligrams of a drug and measure the survival time, you may want to compare the mean survival times of the groups. The downside is that the ANOVA will only tell you if there is a difference, not which groups the difference is between.
A powerful framework for statistical analysis is the linear regression family. By using multiple linear regression, you can analyze large data sets with many variables. For example, let's say you have a data set with demographic statistics, crime rates and police spending. By using multiple linear regression, you would be able to determine if spending on police is correlated with a reduction in crime, with the effects of demographics held constant. The downside of this technique is that some of the data assumptions that have to be met in order to control for statistical error can be onerous.
If you have multiple dependent measures, then you will want to look at multivariate techniques, such as MANOVA or repeated measures MANOVA. The techniques let you test the effect of predictor variables on a set of measures. These techniques are useful when dependent measures are intimately related, or when the relationship between your variables violates the assumptions of the more popular univariate techniques. The drawback of multivariate techniques are their complexity; a strong background in linear algebra and univariate statistics are a must.