We need a book that has a good section on research papers and writing with MLA style. Some grammar would be good too. (A book like this may not exist.)
I am looking for a 101 text, not a style manual or handbook. The text would introduce writing essays in different styles (argument, narrative, process, etc.), and include a chapter on research papers.

YES, there are several. Examples: Writers Handbook (Harbrace)..2ND edition, Writers Inc. Write for College: Student Handbook, or 6th edition, Work in Progress. My fave is the Writers Handbook. Writers Harbrace Handbook has sections on the writing process, MLA Citations, APA Citations, Grammer, researching, diction, effective sentences, mechanics and punctuation. Check your local book store (Barnes/Noble etc.)

I would try A Work In Progress , you can find it on Textbook411.com for a good price, its where I buy all my books

St. Martin Handbook - it covers MLA, APA, and chicago writing style (including how to cite sources that you found). Also, it contains a lot of grammatical rules too, including some references on how to improve your english if you are not a native speaker of english.
I think it includes a section on how to write a research paper too