In my opinion the best foundations to any modern topic in homotopy theory, and derived algebraic geometry in particular, is "Higher topos theory" of Lurie. The scope covers all the required ($\infty$-)categorical framework, and every chapter starts with a very conceptual motivation. In addition, the book also contains appendices which explain classical material (such as model categories) in a very readable way. You might find in the beginning some proofs which involve technical combinatorics of simplices. Don't be discouraged. Feeling comfortable with simplices is essential and this requires working out some details. The proofs in the book do become increasingly conceptual with each chapter, as the concepts themselves get built and acquire depth.