How to solve this regression problem? I have a multivariable regression problem which basically forms a plane in 3D space. There may be other independent variables at play, but if I can approach the problem with 2 independent variables then I should be able to scale to include the rest. 

Broadly, it's a resource problem. I have a required resource (z), which depends on time (x) and number of registered users (y). Registered users is a function of time, but I need the time component because the resource requirement is different at different times of the day. I want to be able to predict the resource requirement at a given time of day if our users continue to registered at a known rate.

Here is a chart of the data:



This is obviously not a linear regression problem, so how do I approach this?
If you want to predict, you need an underlying model of the behavior. Then you can perform least-squares fitting to determine the model parameters. Linear regression just assumes that you have a linear relationship and predicts from that relationship - there's nothing magic about it, except that the model is simple and often effective. Polynomial regression gives bad behavior when extrapolating, so I wouldn't advise that.

Building a model function is usually a trial-and-error process (unless you have some prior knowledge about your system, which I guess you don't). I'd suggest looking at the locus of (x, y) points where z is maximized. It looks like they have an exponential decay relationship. A model like y = exp(-x/x0) might fit the data rather closely. Once you've determined the value of x0 from the data, you can rebuild a model for z.