Suggest books in calculus to improve problem solving skills

If you are looking a book which has plenty of exercises on basic calculus, you could try B.P. Demidovich: Problems in mathematical analysis (originally Demidovich: Zbornik zadac i uprazneni po matematicheskomu analizu, there are translations to several languages). Even Russian edition might be good for you - for computing problems, like calculating a limit, a derivative, an integral you do not need understand the text of exercise. You can probably find it online, e.g. try searching for: demidovich zbornik djvu or Демидович Сборник задач djvu or demidovich problems djvu and similar searches (you can try pdf instead of djvu).

I very much like the selection of exercises in this book.

Kaczor, W. J.; Nowak, M. T. Problems in mathematical analysis. I. Real numbers, sequences and series. Google Books MR1751334

This book has two more volumes (II. Continuity and differentiation and III. Integration). There exists English, French and Polish edition.

Many interesting problems on real analysis are in

Rădulescu, Teodora-Liliana T.; Rădulescu, Vicenţiu D.; Andreescu, Titu. Problems in real analysis. Advanced calculus on the real axis. Google Books, MR2514007.

Although I am not sure whether this is at the level you're aiming for. (But you asked about real analysis in your question, too.)

You might try Bluman, "Problem Book for First Year Calculus", Springer-Verlag 1984.