What are some ways in which an English class and a Linguistics class can be tied together?
English classes (I am assuming you are referring to a typical English literature/grammar class and not an English Language class (meaning that the English language is taught))tend to include the study of both grammar and literature (a combination of short stories, essays, novellas, novels, fiction, and nonfiction). Excellent studies should include a lesson (or multiple lessons) which focuses upon the historical aspects of the period from which the text comes from. This study helps students to understand how the time period influenced the text.
Linguistics, superficially, is the study of language. This complex study examines the rules and representations of language, including grammar (one tie to literature/English), semantics (the study of meaning), syntax (the study of language principles and processes), and aspects of language comprehension and development.
English classes and linguistics are naturally tied based upon the fact that both examine grammar (rules), meaning (when a text is older and a word's definition has changed over time), how authors structure their language (especially when examining poetry or stylistic writing choices), and a teacher's knowledge of how students learn language (when examining vocabulary, spelling, or memorization).