Asking for a good starting tutorial on differential geometry for engineering background student. I just jumped into a project related to an estimation algorithm. It needs to build measures between two distributions. I found a lot of papers in this field required a general idea from differential geometry, which is like a whole new area for me as a linear algebra guy.   I indeed follow a wiki leading studying by looking up the terms, and start to understand some of them, but I found this way of studying is not really good for me, because it is hard to connect these concepts.
For example, I know the meaning for concepts like Manifold, Tangent space, Exponential map, etc.  But I lack the understanding why they are defined in this way and how they are connected.
I indeed want to put as much effort as needed on it, but my project has a quick due time, so I guess I would like to have a set of the minimum concepts I need to learn in order to have some feeling for this field.
So in short,  I really want to know if there is any good reference for beginner level  like me -- for engineering background student?  Also since my background is mainly in linear algebra and statistics, do I have to go through all the materials in geometry and topology ?  
I really appreciate your help.
Following up:  I checked the books suggested by answerers below, they are all very helpful.   Especially I found Introduction to Topological Manifolds suggested by kjetil  is very good for myself.    Also I found http://www.youtube.com/user/ThoughtSpaceZero  is a good complement (easier) resource that can be helpful for checking the basic meanings.  

I suggest you read Lee's introduction to topological manifolds followed by his introduction to smooth manifolds.
