Reference Request: Algebraic Topology and Geometry with External Motivation

I think you're looking for Introduction to Topological Manifolds by Lee: http://www.springer.com/us/book/9781441979391.

A book that approaches algebraic geometry from a more geometric perspective is Principles of Algebraic Geometry by Griffiths and Harris: http://au.wiley.com/WileyCDA/WileyTitle/productCd-0471050598.html.

Motivation for studying much of this stuff in the context of mathematical physics can be found in Topology, Geometry and Gauge Fields by Naber: http://www.springer.com/gp/book/9781441972538.