from Kitzur Shulchan Aruch printed by Moznaim Translated by Eliyahu Touger ch 17 page 153 halacha 7:
Before reciting the Shema, [in the blassing, Ahava rabbah] when reciting the phrase ve'havi'einu ("and bring us [from the four corners of the world"]), a person should take his tzitzis in his hand. During the recitation of the Shema, he should take hold of them in his left hand. During the recitation of the Shema, he should take hold of them in his left hand, between his forefinger and middle finger, opposite his heart.
When he reaches the passage vayomer, the passage concerning the tzitzis, he should hold them in his right hand. When he recites the verse ur'isem oso, he should place them on his eyes, look at them and kiss them. It is customary to kiss them every time one recites the word "tzitzis".
Practically this means that initially he hold the tzitzis just with the left hand and during the passage concerning tzitzis while still holding the base of the tzitzis in the left hand he also takes hold of the tzitzis in his right hand in order to bring them to his eyes and to kiss them. Note in this passage quotes it might sound like a switch but it is not, as it only says to take hold of them implying that one does not release them from his left hand only that he also takes hold of them with his right hand for the third paragraph of Shema.
additions from shulchanharav website footnotes in regards to this matter:
 Arizal in Shaar Hakavanos p. 27b “When one begins Vayomer he is to take hold of the Tzitzis also in his right hand, leaving the roots in the left hand”; Kaf Hachaim 24/8; Ketzos Hashulchan 19/25; Custom of Rebbe as witnessed by congregants; See Sefer Haminhagim p. 24 [English] of the custom of the Rebbe Rashab “They were held [in the left hand] throughout the Shema, including the stages when they were kissed.”
Other opinions: Some Poskim write that at Vayomer one is to hold the Tzitzis only in the right hand. [P”M 24 A”A 1; See Kaf Hachaim ibid that negates this ruling.]