Who is doing the printing? You, or a commercial printer?
What do you mean it needs to start on page 2? Unless you force ID to put pages on the opposite side by turning off page shuffling and renumbering, odd pages will be on the right and even pages on the left for left-to-right documents.
I mean that instead of having a single page dangling at the beginning and the end, the document needs to consist of pairs of pages side-by-side throughout. My professor said that we can achieve this by starting the document on page 2.
This will be done by a commercial printer.
Commercial printers DO NOT WANT you to "impose" your file. That's their job and they know how to do it. If you try to do it for them you just make things more complicated, and it may even make the file unusable.
Assigning page 2, or any other even number, to the first page of the document using the numbering and section options will give you a two page spread to start as your professor has said, but do you really want to start on page 2? The numbers you assign using these options are the numbers that are used for automatic page numbering. In addition, change a page from odd to even, or vice versa, will move it to the other position and wreak havoc on an already laid-out project, as you've discovered.
If you need to impose (arrange the pages for print) to create a sample for your class that you will print out and hand in you should either use File > Print Booklet to print directly to a printer OR use a script to create an imposed copy that you can export to PDF and print.
For simple 2-up saddle stitch (a larg page folded in half, then stapled in the middle) which is what you are doing, my two favorite scripts for imposing are by Dave Saunders, one of our former moderators. if there is no content that crosses over the spine to appear on both pages in a spread you can use the basic script: http://pdsassoc.com/downloads/Buildbooklet.zip
This script will make a copy of your file and rearrange the pages, setting section options to preserve the numbers. You then export to PDF with the Spreads box checked to get your two pages per sheet.
If you have crossovers, you need Dave's CS3 Booklet Script which creates a new file that is twice as wide as the original, then places the pages from the original into it in the correct pattern. You export this file WITHOUT checking the scripts box.
For help on how to install and run a script, see How to install scripts in InDesign | InDesignSecrets
I's been some time since I needed to install either of these scripts, so they may no longer work correctly in the newer versions of ID. If you find that you see script errors when trying to run them, either one should work if put into a subfolder in the Scripts Panel folder named "Version 5.0 Scripts" (without the quotes, but the name is case-sensitive).
I can confirm that printers do NOT want your pages imposed. So there is no need to impose.
However, if you have seperate files for covers and the inner works, you can always start on page 2 as Peter explained.
If you do your own printing from a inktjet or laser printer, you can use the Print booklet option in InDesign.