Is Apple ID required? I just got my first iPhone 6. I wonder if it is a requirement to get an Apple ID or if it is possible to install apps without having an Apple ID? I think that I already have so many IDs and I don't want a new.
Yes, you need to use an Apple ID in order to download/buy apps/music/movies/books.

Apple assigns all transaction like downloaded apps (even "buying" a free app) to your account so you can download them later. One Apple ID is all you need in order to move freely in Apple's ecosystem.

Whenever you sign in on a new iPhone with your Apple ID your previous downloads will be available without the need to buy them again. The same is true for purchased music/movies/books/ringtones etc.

As noted by @SteveChambers you can use your current email address as your Apple ID.