Maybe Introduction to Cosmology by Barbara Ryden might be a good choice.
Assuming second or third year undergraduate physics and math as input, this well written book gives an elementary, but detailed and quantitative, treatment of the expansion of the universe, dark matter, dark energy, inflation, big bang nucleosynthesis, and structure formation (galaxies and superclusters of galaxies). Knowledge of general relativity is not a prerequisite. The FLRW metric is used and studied, but is not derived.