I only have so much to work from my memory but here it goes:
The story was about a normal child that was pulled into a fictional world where Wyverns stalked the land. The protagonist (a youngish boy I believe) comes into contact with a dragon that is not full grown. Over the course of the series the dragon grows as they quest throughout the land.
The series would have been around in the early 1990s.
I also seem to think that the titles had common "punnish" titles but am unsure on that point.
I have tried to find these books for years as they were an entry point for me into fantasy literature and were a staple of my childhood library forays.
I do recall an additional detail from the first book. They are in a village or town and it is dark during one scene in which they have to remain inside for fear the wyverns will locate and burn them. The wyverns are like smaller dragons with the ability to breathe fire.