Question:

I've recently started wrapping tefillin every morning and saying the Shema. Eventually, I hope to build from there, especially once I move to a Jewish community where this is a daily minyan.

Like I said, I'm doing this every day. Now I'm told that I'm supposed to skip Shabbat mornings. Are there any other days to skip—and why?

Response:

Great to hear you've made this a habit. But even the best of habits needs a break once in a while. So here are the rules:

Jewish men (over the age of 13) wrap tefillin every day other than the holidays listed below:

Saturdays (Shabbat)
Rosh HaShanah
Yom Kippur
Sukkot
Shemini Atzeret/Simchat Torah
Passover
Shavuot

Concerning the intermediate days of Sukkot and Passover (chol hamoed), there are a number of divergent customs. Sephardim, Chassidim (including Chabad), and many Ashkenazim do not wear tefillin, while other Ashkenazim do. Among those who do wear tefillin, there are some who recite the accompanying blessings and some who do not. (Have a look at Do I put on tefillin on Chol Hamoed? for more on this.) Make sure to speak to your rabbi to ascertain what your community's practice is.

Now, you may ask, if these days are so special, shouldn't they be special for wrapping tefillin as well?

So here's the explanation taught by Rabbi Akiva, cited in the Talmud Menachot: The Torah (Exodus 13:9) refers to tefillin as a sign of the special bond between G‑d and the Jewish people. Shabbat and the holidays are also a sign (Exodus 31:13) of this bond. Since there already is the special sign of the holiday, the additional sign of the tefillin is superfluous. As rabbis and architects are wont to say, more is less. Wearing tefillin on those days would demean the sign of the holiday.

Also note, on the fast of Tisha B'Av, the donning of talit and tefillin is delayed until the afternoon, as a mark of mourning.

Here is a chart for reference:

Day/Holiday Do I Wrap Tefillin?
Sundays-Fridays Yes
Saturdays No
Rosh HaShanah No
Yom Kippur No
First and final days of Sukkot No
Intermediate days of Sukkot Depends on custom
Minor fast days Yes
Chanukah Yes
Purim Yes
First and final days of Passover No
Intermediate days of Passover Depends on custom
Lag BaOmer Yes
Shavuot No
Tisha B'Av Delayed until afternoon

Your truly,

Rabbi Menachem Posner

Sources
Talmud, Menachot, 36b.
Code of Jewish Law, O.C. 31, 555.