English Translation
One day I strolled into the woods.
過去のある日 sounds very unnatural. ある日 or とある日 is enough for “one day”.
ぶらぶらと歩いて森の中に入る is also unnatural for “to stroll in the woods”. I would translate:
私はある日、森を散策した。（わたしはあるひ、もりをさんさくした。）
散策する means “take a walk around a certain place with no particular purpose, just for relaxing and enjoying nature.”
散歩する is a synonym of 散策する, but 散策する is better for the woods where you can find many beautiful trees, flowers, animals. The Kanji 策 has the connotation of “to explore”.