The Shulhan Aruch writes (Orah Haim 28; listen to audio recording for precise citation) that one is required to touch his Tefillin “at every moment” while he wears them, so that his mind will not be distracted from them. The Mishna Berura (commentary by Rav Yisrael Kagan, 1839-1933) clarifies that this refers to every time when a person mentions the Tefillin in the prayer service. In the Shema recitation, for example, when one recites “U’kshartam Le’ot Al Yadecha,” which refers to the Tefillin Shel Yad, he should touch the Tefillin Shel Yad, and when he recites the next phrase – “Ve’hayu Le’totafot Ben Enecha” – which refers to the Tefillin Shel Rosh, he should touch the Tefillin Shel Rosh.
The Shulhan Aruch emphasizes that whenever one touches his Tefillin, he should first touch the Tefillin Shel Yad, and then the Tefillin Shel Rosh. The Mishna Berura explains that this Halacha is based on the principle of “En Ma’abirin Al Ha’misvot,” which means that one must never skip over the opportunity to perform a Misva. Since one’s hand in its normal position is closer to the Tefillin Shel Yad than to the Tefillin Shel Rosh, one must first touch the Shel Yad, for otherwise he will be “skipping” the Shel Yad to touch the Shel Rosh.
The Mishna Berura comments further than besides serving to keep one’s mind focused on the Tefillin, occasionally touching the Tefillin is important as a means of checking to ensure that it is positioned properly. If one feels the Tefillin Shel Yad or Tefillin Shel Rosh and notices that it has shifted from its proper location, he must immediately return the Tefillin to its correct spot. If one notices that both the Shel Yad and the Shel Rosh are out of position, then he should first return the Shel Yad to its place, and then the Shel Rosh. The Torah (in the aforementioned verse in the Shema section) first mentions the Shel Yad before the Shel Rosh, indicating that one must place the Shel Yad before donning the Shel Rosh. By the same token, then, if they are both out of position, one should first move the Shel Yad and then the Shel Rosh.
When one removes his Tefillin, he should first remove the Shel Rosh before removing the Shel Yad. The Torah speaks of the Tefillin Shel Rosh in the plural form (“Totafot”), indicating that the Tefillin Shel Rosh is worn when both Tefillin are worn, meaning, when the Tefillin Shel Yad is also worn. Thus, one should avoid a situation where the Tefillin Shel Rosh is on his head but the Tefillin Shel Yad is not on his arm. For this reason, one must remove the Shel Rosh before removing the Shel Yad.
One should stand while removing the Tefillin Shel Rosh.
The Mishna Berura writes that it is proper to remove the Tefillin Shel Rosh with one’s weaker hand. A right-handed person should thus remove the Tefillin Shel Rosh with his left hand, and a left-handed person should remove his Tefillin Shel Rosh with his right hand. The reason for this practice is to demonstrate that we are not enthusiastic about removing our Tefillin, and we therefore do so with our weaker hand, which moves with less strength and vigor.
Whenever one handles his Tefillin, he should hold it with both hands, carefully and delicately, the way one holds an infant. He should hold the Bayit (box of the Tefillin) with one hand, and the straps with the other, so that they don’t dangle. This is the proper way to hold the Tefillin as an expression of honor and reverence. Likewise, while removing the Tefillin Shel Rosh, one should use one hand to remove the Bayit (a right-handed uses his left hand, as discussed above) and the other to hold the straps. Similarly, when one removes the Tefillin from their bag, he should remove them with both hands, and not allow the straps to drag or dangle.
The Mishna Berura records a custom to kiss the Tefillin both when putting them on and when removing them. Furthermore, the Mishna Berura writes, one should not wrap the Tefillin straps on the Bayit; he should instead wrap them on the Titura (the base of the Tefillin). However, this Halacha is not generally applicable nowadays, since we commonly keep the Tefillin in a protective plastic box, and it is certainly permissible to wrap the Tefillin straps on the box.
The Mishna Berura also records a custom to wrap the Tefillin in the shape of wings, to commemorate the incident recorded in the Gemara where a man’s Tefillinmiraculously turned into dove’s wings.
Summary: It is proper to touch one’s Tefillin at various points in the prayer service; one should first touch the Tefillin Shel Yad, followed by the Shel Rosh. The TefillinShel Yad is always put on before the Tefillin Shel Rosh, and the Tefillin Shel Rosh is always removed before the Tefillin Shel Yad. It is proper to use one’s weaker hand when removing the Tefillin Shel Rosh. In general, whenever one handles his Tefilin, he should hold them delicately, with two hands, and ensure that the straps do not drag on the floor or dangle in the air.