Section is, in my experience, the conventional, everyday term for the different physical parts of a newspaper, excluding the magazine, advertising inserts, and other components that are not printed at the same size. Page or pages as Andrew suggests are also in use, as well as feature, but I think of these as parts of the newspaper within a section, rather than as discrete components of it. The science pages might be two pages with a "Science" heading within the main news section, the religion page might be a single article in the lifestyle section.
What is idiomatic will likely depend on what kind of newspapers you are accustomed to reading. It is mostly broadsheets that are traditionally split up into different areas of focus, whereas the division is less useful and less common in tabloid-sized publications, although the latter may have a central section that can be pulled out. Growing up in the 1980s mostly reading the Los Angeles Times and the Orange County Register, I would ask for the business section or opinion section; on weekdays, when there was no separate opinion section, I would ask for the news section or front page section because the editorials were printed at the back of it, where I could find the opinion pages.
Macmillan's first definition states
1a. a part of a newspaper, book, or other piece of writing that may be considered separately
The story was reported on the front page of the business section.
The sports section is probably the most universal, and indeed has its own OED sub-entry with three examples:
- 1912 Oakland (Calif.) Tribune 15 June 1/1 A well-edited sports section, contributed to by experts.
- 1940 G. Marx Let. in G. Marx et al. Groucho Lett. (1967) 46 I picked up the paper Tuesday morning, nervously turned to the sports section.
- 2004 Guardian 12 Jan. (Media section) 8/1 We expect not just a newspaper, but also a weekend section, an arts section, a blow-by-blow sports section, [etc.].
For examples of other kinds of sections, you might consult guides from the Boston Globe and The Washington Post.