I think it is possible to go quite a few steps towards an understanding of stacks, and even that of elliptic curves, without mastering lots of Algebraic Geometry:
There is a wonderful, friendly written, undergraduate readable book which starts out introducing moduli spaces (and later gives a rough idea abut stacks as well): It is Kock/Vainsencher: An Invitation to Quantum Cohomology
Chapter 1-2, maybe 3, can give you a feeling of how you can work with moduli objects, and are a pleasant reading experience. You will get to know moduli stacks/spaces of curves, close enough to elliptic curves for a start, maybe (those latter are trickier to compactify, but first first you have to get far enough to want to to do that, and this book is wonderful for that purpose).
A very readable and short introductory source for Algebraic Stacks is then Gomez: Algebraic Stacks
You say you have no background in Algebraic Geometry - maybe you should know that the concept of stack is not limited to the world of Algebraic Geometry. If you are more comfortable with Topology or Differential Geometry, maybe this helps: Heinloth: Some notes on Differentiable Stacks
If you are categorically minded and can read french, there is a very nice master course on Stacks by Bertrand Toen. It is very abstract - you can fill in your favourite geometric context - and will leave a considerable gap to the study of the concrete moduli stack of elliptic curves, but it explains very well the other, non-moduli, motivations for introducing stacks and algebraic spaces, e.g. "bad quotients". The good thing about is that you won't feel a lack of algebro-geometric background, even "scheme" is defined - but you need categorical background.
Finally, again thinking of your non-Algebraic-Geometry background, the moduli stack of elliptic curves is used in Algebraic Topology (maybe this is your motivation to study it?) and there are some introductions (quite high-level though) aimed at people with an according background. You can look around here
Have fun exploring this nice topic!!