Techelet is a complex issue, and I don’t claim to be an authority, but I can offer my customers some initial guidance.
In recent decades two vanguard organizations claim to have rediscovered the techelet dye referred to in Bamidbar 15:38. The first, Radzyn, is derived from the cuttlefish. The second, Ptil Tekhelet, is derived from Murex trunculus.
Let me warn you that the Radzyn techelet dye runs. Your hands will turn blue, your tallit may get a bit smudged and the white tzitzits will get slightly discolored. From my experience the smudges come out with a damp cloth. The proponents claim this is a sign that the Radzyn dye is authentic, citing an early source that techelet was known to run.
Whether you choose thick or thin is a a matter of aesthetics. The thick may be easier to tie.
If you order Ptil Tekhelet, before you order you’ll need to decide how you intend to tie the tzitzits, because the Rambam set enables you to tie the tzitzits so that one of the eight strings hanging from each corner is blue, while with the Raavad set you get two blue on each corner.
The Ptil Tekhelet website has a wealth of information on the various tying customs. Some of them are quite complicated, while others are fairly straightforward.
The easiest way to tie techelet is to tie it just like white tzitzit strings, using the blue string as the shamash that wraps around the main cord/bundle. If you do this, be sure the very first and very last windings are white, as the Gemara states explicitly.
Further reading on techelet
History, Mesorah, Nignaz by Mois Navon
Ptil Tekhelet on Radzyn techelet by Mois Navon
False Techelet by Mois Navon
Tekhelet – Threads of Reason by Mois Navon
Has Techelet Been Found? by Menachem Epstein