I think graded readers designed for adult learners of Japanese are one of the best ways to train your reading comprehension skills.
This thread is a good place to learn more about graded readers and ask questions about them:
Books written for Japanese children can be difficult material for us language learners because even a native 1st grader knows more grammar and vocabulary than a beginning learner. Plus, they might use casual or childish language that is different from what is taught in our textbooks at a beginning level.