The issue of wearing techelet tzitzit is complex, and I make no claim of authority, however I can offer my customers some basic guidance.
During the past few decades two vanguard organizations claim to have rediscovered the techelet tzitzit dye mentioned to in Bamidbar 15:38. The first, Ptil Tekhelet, is derived from Murex trunculus. The second, Radzyn, is derived from the cuttlefish. (We also use Murex strings made by Techeiles Chabura.)
Today, most Orthodox Jews who wear techelet insist on Ptil Tekhelet, though Chassidim – notably Breslov and of course Radzyn – wear the Radzyn tzitzit. In recent years, even some Breslov chassidim have switched from Radzyn Techelet to Ptil Tekhelet.
Be warned that the Radzyn techelet dye tends to run. If you tie the tzitzit yourself, both your hands and the edges of the tallit may get a bit smudged and the white tzitzit strings will become slightly discolored. From my experience the smudges can be easily removed with a damp cloth.
Whether you choose thin techelet tzitzit or thick is a matter of aesthetics. The thick blue tzitzit may be easier to tie.
If you choose Ptil Tekhelet, before you order you’ll have to decide how you intend to tie the tzitzit, because when tied, with the Rambam set one of the eight strings hanging from each corner is blue, while with the Raavad set you will have two blue strings on every corner.
The Ptil Tekhelet Foundation website presents a wealth of information on the various techelet tying customs. Some of these are quite complicated, while others are relatively straightforward.
The simplest way to tie techelet tzitzit is to tie them just like white tzitzit strings, using the blue string as the shamash that wraps around the main cord. If you do this, be sure the very first and very last windings are white, as explicitly required by the Gemara.
Further reading on techelet:
History, Mesorah, Nignaz by Mois Navon
Ptil Tekhelet on Radzyn techelet by Mois Navon
Fake Techelet - forum string at Mi Yodeya
Tekhelet – Threads of Reason by Mois Navon
Understanding the Criteria for the Chilazon by Dr. Mendel Singer
Response to Understanding the Criteria for the Chilazon by Dr. Baruch Sternman
Has Techelet Been Found? by Menachem Epstein