Principles of cosmology and gravitation by Michael V Berry is a short undergraduate textbook which was originally published in 1976. A new edition was published in 1989, but with minimal changes, so by now the book is looking somewhat dated - it doesn't deal with inflation for instance. But it would still be useful as an introduction to cosmology as it shows that many calculations can be carried out using mathematics which isn't too challenging for a science undergraduate - it doesn't venture into tensor calculus.
The book starts by looking at how cosmological distances are measured (Unfortunately it suffers from the usual Hubble sphere/particle horizon confusion.). Berry goes on to look at why general relativity is necessary for the study of gravitation, and this is followed by an introduction to general relativistic mathematics, leading to the Schwarzchild solution and so black holes. The later part of the book looks at topics in cosmology, with chapters on the kinematics and dynamics of cosmological models, finishing with a chapter on topics such as the CMBR and the development of inhomegeneities in the universe. If you want a cosmology book where it is fairly straightforward to work through the mathematics, and aren't too bothered by the lack of modern topics, then you could give this book a try.
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