For a given miRNA-target interaction (predicted by a tool like targetScan), you could create a scatter plot between paired measurements (which should hopefully have a negative correlation; otherwise, I wouldn't consider this a plausible interaction in your own dataset).

I doubt it that you will see correlation between miRNAs and their target mRNAs since they have very little influence on mRNA levels. You should expect to see similar patterns of miRNAs and mRNAs in similar cells (or cultures, conditions or whatever you're looking at) but this will tell you more about the TFs that set the pattern of the mRNAs and miRNAs.

An intuitive way of combining the data would be to rank the miRNAs in each sample according to the expression and rank the miRNAs in the sample and then you can plot both of them on the same figure. You can then try and do biclustering (e.g. Tanay et al: http://bioinformatics.oxfordjournals.org/content/18/suppl_1/S136.short) to discover sets of miRNAs and mRNAs that determine a specific or a set of conditions.

If you don't want to use ranking and you assume the data in both datasets is normally distributed you can take the STD from the mean as the expression level in each dataset and they will be comparable between the two.