Prior to saying the blessing over the Tzitzis one is required to separate the strings of the Tzitzis from each other. [This applies for both the Tallis Gadol and the Tallis Katan. This applies even if one is not reciting a blessing over the Tallis.]
If one is pressed for time: If one is pressed for time to go to Shul, and if he waits until he untangles the Tzitzis strings he will be unable to pray with the Minyan, then he does not need to separate the Tzitzis. [Some however rule that this only applies if one had separated the Tzitzis the previous day, in which case in a time of need he may rely on that perhaps they have remained untangled. If however one did not do so, or if one sees that the Tzitzis are tangled, then he must untangle the Tzitzis even in a time of need.]
In addition to examining the Kashrus of the Tzitzis, one is also to untangle the strings, as the Torah states Tzitzis, which implies individual strings. In a time of need it is permitted to skip the examination and separation process, such as if one is pressed for time to say Shema with Tzitzis or pray with a Minyan.
Is one to separate the strings of his Tallis also on Shabbos prior to the blessing?
Some Poskim rule one may not separate the strings on Shabbos, [even if they are loosely entangled]. Others however rule that one may separate [loosely] entangled strings on Shabbos. Practically, one is to do so, although one must beware not to untangle any strong knots that may have formed, and also not to tighten the top knot of the Tzitzis which commonly becomes loose. [As a precaution to prevent tightening the upper knot on Shabbos, upon checking the Tzitzis strings on Shabbos, one is to hold the upper knot of the Tzitzis and then separate each string.]
A new Tallis: If it is a new Tallis and it is the first time one is wearing it, then one is not to separate the strings at all on Shabbos, and is rather to do so on Erev Shabbos.
Does one fulfill the Mitzvah in the event that he wore Tzitzis with tangled strings?
May one wear Tzitzis if the strings are tied or glued to each other in a way that they cannot be untangled?
Some write that if the strings cannot be untangled then the Tzitzis is invalid. Seemingly however the Tzitzis remains valid so long as the top of each string is loose and extends beyond the area of entanglement.
Being careful to constantly untangle the Tzitzis:
The Arizal stated that the acronym of the word Tzaddik is “Tzaddik Yafrid Tzitziyosav Tamid”, that a Tzaddik constantly separates the fringes of his Tzitzis.
 Admur 8/12; Siddur Admur; Michaber 8/7; Menachos 42a
 The reason: The reason for this is because the Torah calls the fringes “Tzitzis”, and the term Tzitzis refers to separated strings, similar to hairs that are separated from each other, as it says in the verse [Yechezkal 8:3] “And he takes me by the Tzitzis of my head” [Admur ibid; Tur; Levush] Other reasons: Other Poskim however write that today this is done in mere memory of the Techeiles, as in times of the Techeiles, from the letter of the law one was required to separate the Techeiles from the Lavan. [M”A 8/10 in name of Baal Haitur in Menachos ibid] Some Poskim however argue on the M”A ibid and rule the separating is required from the letter of the law. [Biur Hagra; Olas Tamid, brought in Biur Halacha 8/7 “Tzarich Lehafrid”]
Must one untangle the entire string? Some Poskim question whether it is necessary to untangle the strings prior to the blessing, if the top of the strings are loose and extend beyond the entangled area, as it is valid to have a majority braid and minority loose string [see 11/26], and thus it should suffice to have even a small amount of string protruding from the entanglement. [Ketzos Hashulchan 7 footnote 6]
Must one untangle each of the 8 strings from each other? See Ishkavta Direbbe footnote 16 [p. 21] that according to the M”A ibid one is only required to separate the strings of one side from the other side, however not from each other. However according to the Gr”a, Olas Tamid, and simple implication of Admur, all the strings must be untangled.
 Ketzos Hashulchan 7/7; Igros Kodesh 6/202; 7/348
 Pashut! See the previous Halacha regarding checking the Tzitzis that Admur 8/14 rules it applies even if a blessing is not recited.
 Admur ibid; Siddur ibid; M”A 8/10; Chayeh Adam 11/41; Elya Raba 8/8; Kitzur SHU”A 9/7; Derech Hachaim; M”B 8/18 [however see Biur Halacha]; Aruch Hashulchan 8/13; Ketzos Hashulchan 7/3; Chazon Ish 3/9; Poskim in Kaf Hachaim 8/29
Other opinions: Some Poskim rule that the Tzitzis must be separated from the letter of the law, and thus perhaps if one did not do so, one may not say a blessing over wearing it. [Biur Halacha ibid based on Gr”a ibid and Olas Tamid] To note that although Admur ibid implies the untangling is required from the letter of the law [and omits the reason of the M”A] nevertheless he brings the ruling of the M”A in this regard.
 The reason: This is due to the fact that untangling the Tzitzis is a mere stringency in memory of the Techeiles. [M”A ibid] Others however learn that according to Admur this is due to that one may rely on a Chazaka that it was untangled the day before, as rules Admur in 8/15 regarding skipping the examination process in a time of need. [Ishkavta Direbbe p. 21]
Does this allowance apply even if he sees that the strings are tangled? Some write that according to Admur the allowance only applies if one had separated the Tzitzis the previous day, in which case in a time of need he may rely on that they have remained untangled. If however one did not do so, or if one now sees that the Tzitzis are tangled, then he must untangle the Tzitzis even in a time of need. [Ishkavta Direbbe p. 21] Nevertheless, this is not implied from Admur which writes “he does not need to untangle them” hence implying that this applies even if they are definitely tangled.
 Ishkavta Direbbe p. 21 [see previous footnote], based on the explanation of Admur in 8/15, that the examination process may be skipped since one knows that it was valid the previous day; See Tzitzis Halacha Lemaaseh p. 128.
 Gan Hamelech 65; Kesher Gudal 2/3; Shalmei Tzibur 29; Shesilei Zeisim 8/13; Chesed Lealafim 8/8; Shaareiy Teshuvah 8/8; Kisei Eliyahu 8/5; Ben Ish Chaiy Bereishis 3; Kaf Hachaim 8/30; Artzos Hachaim 8/7
 The reason: This is forbidden due to a possible Shabbos prohibition of fixing a vessel. [ibid]
 Implication of Mateh Moshe 443; Kneses Hagedola 281/4; Kisei Eliyahu; Yifei Laleiv 8/16; Chayeh Adam 8/17; Mamar Mordechai 8/9 brought in Kaf Hachaim ibid; Shulchan Hatahor 8/5; Halichos Shlomo 3/5; Yabia Omer 5/3
 So is implied from Admur 8/13 in Kuntrus Achron 2
 See Shulchan Aruch chapter 317 for the laws of tying and untying on Shabbos; Ketzos Hashulchan 123 footnote 4; SSH”K 15/7; Az Nidbaru 3/22; Toldas Shmuel Kosher 9; Piskeiy Teshuvos 13 footnote 17
 Kovetz Oholei Lubavitch 3/5 that so showed the Rebbe to a young married student the order of checking on Shabbos. [printed in Shulchan Menachem 1/40]
 See Poskim ibid of lenient opinion that when the strings are being untangled for the first time it is possibly Tikkun Mana. [Mamar Mordechai ibid in his explanation of the Gan Hamelech]
 Admur 8/12 and all the other Poskim ibid that rule in a time of need he allows to wear the Tzitzis without separating the strings; This is unlike the understanding of the Biur Halacha 8/7 “Tzarich Lehafrid” that according to some Poskim perhaps one does not fulfill his obligation. See also Admur 23/3 “Some are accustomed upon entering into a cemetery to tie the Tzitzis of two corners to each other in order to nullify the Mitzvah of Tzitzis from them [and hence allow one to enter the cemetery with the Tzitzis revealed]. However, in truth this does not help at all, as although the Tzitzis are tied to each other, nevertheless the Mitzvah of Tzitzis is not nullified through doing so, as he intends to return and untie them. [Admur 23/3; M”A 23/1]
 Piskeiy Teshuvos 8/17 based on Chazon Ish 3/9
 See Ketzos Hashulchan 7 footnote 6 and Admur 11/26 that there is no Shiur for the Anaf, and hence so long as there are a valid amount of strings that extend past the entanglement it remains Kosher.
 Elya Raba 8/8; P”M 8 A”A 10; M”B 8/18; Kaf Hachaim 8/29