It is up to you how you want to print and bind the songbooks but over time I've learned a few things.

Paper

While standard office paper has 80 g/m², most printers and copiers could handle paper as light as 60 g/m² which would make the songbooks thinner and 25% lighter.

Binding the books

Ring binders turned out to be too bulky and heavy to cary around. The songbooks are intended to be used at campfires and the likes and should be lightweight and easy to use. File folders are lightweight, but flipping pages is very cumbersome.

I ended up using book rings, as these are very lightweight, cheap, and allow for very easy page flipping. It's also very easy to open the rings and swap out or insert pages if you want to update your songbook. Book rings are sometimes hard to find in stores, but in Vienna you can get them at Gerstaecker and Mastnak (they are quite expensive there though). I wasn't able to find any at Libro, Müller or Pagro. You can also get them at staples or Amazon.

Front and back covers

As a back cover you could use mattes or cardboard and I always print the title page on thicker paper (120 g/m²) to make it more sturdy.