Both, and more, but it's up to the user to decide how to use their screen reader!
Screen readers provide a variety of ways to move around the content. There's no need for designers, developers, or authors to try to manage this on the user's behalf. The important thing is to ensure the content is well organized (using semantic markup) so that users can digest the content using whichever of the screen reader's features suits them.
I am designing a web application, which can be accessed by any visually impaired person. This web application contains various pages, and each page can have paragraphs with links, form input fields, collapsible sections and buttons.
The key word here is "paragraphs". It indicates you have some non-operable content on the page. Links, buttons, and form inputs are operable; you can make them do something. Paragraphs, sentences, and headings aren't operable; all you can do is read them. Screen readers provide different ways to move around operable and non-operable content.
When my user opens the page, should screen readers read the content on my page automatically, element by element, from top to bottom?
That's entirely up to the user. Screen readers can start reading a page automatically, or wait until the user asks for it.
They can also read various sized chunks one at a time: block, line, word, or character. Typically they use the arrow keys or a touchscreen swipe per chunk. This way users can read the content at a pace which suits them, or go back and read something again. Or they can keep reading everything until the user says stop.
Confident screen reader users will often swap between these different approaches on the fly. For example, a user might switch to reading one character at a time in order to clarify a phone number, or disambiguate words which are homophones.
Or, should users take control by using tab key and navigating through the page, having the screen reader read the content where the cursor focus is?
The important thing to understand about the TAB key, is that it is used to move from one operable item to the next. Any non-operable content in between will be ignored. So if a paragraph has some links inside a sentence, pressing TAB will just read out the links, not the rest of the sentence. The TAB key also stops at buttons and form inputs.
When the user wants to read the whole paragraph, they will proceed using the arrow keys. When dealing with a form or application-like UI, they can use the TAB key.
These aren't the only navigation methods offered by screen readers though. It's also possible to jump from one heading to the next, or between controls of a given type (e.g. "jump to next checkbox"). There are also commands for exploring data tables by row or column.
Suggested reading: Responses To The Screen Reader Strategy Survey. Here a group of screen reader users describe the various ways they use their tools to navigate and understand how a page is organized.
