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?
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:
Both days of Rosh HaShanah
The first and last two days of Sukkot
The first and last two days of Passover
Both days of Shavuot
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.
Concerning the intermediate days of Sukkot and Passover, there are a number of divergent customs. Sephardim and some Ashkenazim (including Chassidim) do not wear tefillin, while others 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.
In addition, on the fast of Tisha B'Av, the donning of talit and tefillin is delayed until the afternoon, as a mark of mourning.
Rabbi Menachem Posner
Talmud, Menachot, 36b.
Code of Jewish Law, O.C. 31, 555.