People who attend synagogue on the first and last days of Pesach, and the intermediate Shabbat, know that there are special Torah-readings for those days. The regular weekly schedule of parshiyot is temporarily “sidestepped.”
Torah is also read on every day in-between (as it is on Sukkot and Chanukah — the other 8-day festivals on the Jewish calendar) the first and last days (where ordinarily, it would only be read on Monday and Thursday mornings). Reading Torah on these days helps maintain the special spirituality — the “holiness” — of the days, by maintaining meditation on the holiday-themes, even though the days are “hal ha-mo’eid” (intermediate days), rather than “chagim” (holidays) themselves.
Below, I’ve taken the list of Torah readings for intermediate days, and their description, from Chabad.org. 
Anyone reading this piece in any year can check the same website for the schedule of readings in that year.
I’ve gleaned some related comments from various commentaries and anthologies. As part of your own, private meditative learning, why not search out more comments on these Torah-sections, or on specific verses within them?
1) Exodus 13:1-16: Instructions to commemorate the Exodus by sanctifying the firstborn, avoiding leaven and eating matzah on Passover, telling one’s children the story of the Exodus, and donning tefillin.
13:3 — [זכר] — “Remember. This word is in the infinitive form, implying [in Hebrew grammar] that the Exodus should be remembered constantly.” 
13:9 — For a sign to you…: “The Exodus is to be more than a mere annual celebration. Its eternal lessons are to be ever before the mind of the Israelite…” 
2) Exodus 22:24-23:19: A portion from parshah “Mishpatim,” which includes the laws of the festivals.
“Each festival [Pesach/Sh’vuot/Sukkot] is called a chag, a term that indicates that the festival involves an obligatory pilgrimage to a sanctuary [i.e. the Temple in Jerusalem]. The same meaning endures in the Muslim institution of the chajj [often pronounced hajj], the religious duty to make a pilgrimage to Mecca [at least once in a lifetime].” 
23:14 — Three times…: “They were to rejoice in their Maker.” 
3) Exodus 34:1-26 (note — when one of the “intermediate days” of Passover is Shabbat, this is the reading read on that day, and it begins 12 verses earlier, with 33:12): A section describing Moses’ receiving of the Second Tablets and G-d’s revelation to him of His Thirteen Attributes of Mercy, which likewise concludes with the laws of the festivals.
34:6 — Kindness and truth…: “Lovingkindness precedes truth, both here and in Scripture, as if to say, ‘Speak the truth by all means, but be quite sure that you speak the truth in love’.” 
4) Numbers 9:1-14: The story and laws of the “Second Passover.”
9:1 — And the L-rd spoke to Mosheh in the wilderness of Sinai…: “Why was Torah given in the wilderness? To teach you that whoever doesn’t make himself [or herself] free to all as the wilderness [i.e. open; humble] is not worthy to receive Torah…” 
 “Stone” Chumash; Art Scroll Publications; p. 361 (based on Rashi)
 Hertz, Rabbi Joseph H.; Pentateuch and Haftorahs; p. 261
 The Rabbinical Assembly (Conservative); Etz Hayyim; Torah and Commentary; p. 473
“The Arabic letters are ‘hah’ and ‘jeem’. The ‘hah’ has the same sound as ‘chet’ and the ‘jeem’ sounds like the ‘j’ in Jew in Standard Arabic [i.e. soft ‘g’], though in Egyptian Arabic it sounds like ‘gimel’ [i.e. hard ‘g’.]” (received in private email from linguist Simon Ager 4/23/11).