To continue reading can be phrased as "to keep reading" or "to continue to read." Here are a few examples of how you could use these phrases:

"I'm going to try to keep reading for at least another hour before I take a break."
"I'm enjoying this book so much that I can't wait to continue reading it tonight."
"I find that reading helps me relax, so I try to continue reading a pttle bit every day."