This is my first time to plan a trip. I'm going to visit Amsterdam and then go to Paris. Will my single-entry Schengen tourist visa allow for my planned trip?
Of course, the Schengen Area works as a single country for border purposes. A single entry visa allows for one entry into the area, and you're free to move within it during the validity period of your visa.
Your passport will not normally be checked at the Dutch-Belgian or Belgian-French border. If it happens, it's just an internal check to see that everyone's legally in Europe, and no stamps are given.
Yes, it will be permitted. Your passport will not be checked on your journey from Amsterdam to Paris. (Sometimes, there can be a check for drugs on the border)