Based on the ruling of Rav Nahman in the Gemara (Menahot 39a), the Shulhan Aruch (O.H. 9:1) says that one is Biblicallty obligated to have Tzitziot (fringes) on a four-cornered garment only if the garment is made of wool or linen. If a four-cornered garment is made of other materials, such as cotton or silk, one is required to have Tzitziot only on a Rabbinic level. This is also the opinion of the Rif and the Rambam. However, the Rama cites another opinion in Menahot and rules that a four-cornered garment of any material requires Tzitziot on a Biblical level, and this is a major difference of opinion between the Ashkenazic and Sephardic communities. Despite the Rama’s opinion, the Mishna Berura (O.H. 9:5) does state that one should strive to wear a wool Talit or Talit Katan in order to fulfill all opinions.
Practically speaking, one may wear a Talit Katan or Talit Katan T-shirt made of cotton but according to the the Sephardic approach, one would be fulfilling a Rabbinic Mitzvah only. Regarding the Talit Katan which is made of netting or mesh material, Rabbi Yisroel Belsky (Shulhan HaLevi) rules that one does not fulfill the Mtizvah of Tzitzit because the whole premise of a garment is something that covers and protects. Mesh material is mostly open space and as such does not fulfill the requirement of a garment. If possible one should attempt to avoid wearing such a Talit Katan. Furthermore, if one cannot wear a woolTalit Katan everyday, one can try and wear it at least on Shabbat to merit the Mitzvah on a Torah level.
The Shulhan Aruch (O.H. 9:5), based on the Rambam (Hilchot Tzitzit, 2:8), also says that it is proper that the Tzitziot of the Talit be the same color as the Talititself. Regarding Talitot that have black or blue stripes, the Pri Megadim says that that does not detract from this Halacha since the majority of the Talit is white, just like the Tzitziot. Although it is common to see people in the Sephardic community wearing all-white Talitot or Talitot with white stripes in order to conform to the Shulhan Aruch, this is an considered simply an enhancement of the Mitzvah and not something that is required according to the letter of the law. Indeed many sages in the Moroccan community were not particular of having a totally white Talit and it is worthy to note that the Talit of the Baba Sali had black stripes a well.
Summary: One fulfills the Mitzvah of Tzitzit on a Torah level with a wool or linen garment, and on a Rabbinic level with other materials. One may wear a Talit with different colored stripes, as long as the majority of it is white.