I'm going to take a different approach to Jeromy's answer. The first, and most important, thing to do here is plot the data.
Look at scatterplots of all your independent variables against your dependent variables. If you use R, the
pairs.panels() function in the
psych package will give you a lot of useful information.
Once you've looked at scatterplots, you'll have a better idea of how your variables relate. I've lost track of the amount of times that model results which seemed confusing made a lot more sense when I plotted my results.
I highly recommend ggplot2 (again if you are using R), you can use the
qplot() function, which allows you to add 3 or more variables to a scatterplot by altering colour, shape and size of points based on the levels of another variable.
Then, and only then, should you model your data (using something like regression). Then plot new variables based on the results of your model. This interplay between plotting and modelling is key to how I (and I suspect others) create good models.
In terms of your two dependent variables, I would first plot them against one another to examine if they are highly correlated. If they are, then you can probably create one model with whichever you prefer (but you should replicate this model with the other to be sure). If they are not, then create two seperate models for each of your dependents, and graph them to look for outliers or violations of the model assumptions. While you could use canonical correlations or MANOVA's I would probably build separate models first and then perhaps combine them into one multivariate model, all going well.
If you're not using R, I highly recommend that you start, if only for the wonderful ghraphics possible. Jeromy's blog has some really good resources to get you started if you're from a social sciences background (indeed that's how I got started).
Best of luck, and welcome to the site.