All temple laces are made with white cotton thread. I personally prefer white thread to colored thread anyway, so all my thread supplies are blasting, gleaming, white. The best suppliers for lace thread are (in my never to be humble opinion) DMC Cordonnet Special, and Altin Basak (a Turkish thread supplier). DMC Cordonnet can be ordered online, and the color always matches. Turkish threads are a brighter white, and less expensive, but they’re harder to find. There are other threads too that are useful.
Altar cloth guidelines ask for size 20 or size 30 thread. Lace threads go as fine as size 100, which is finer than sewing thread. The higher the number, the smaller the thread.
All of my temple laces are crocheted rather than tatted, but the temple accepts both crochet and tatted laces for their altars.
Crochet hooks for lace nearly always have to be ordered online. Boye brand hooks work well, I also like Tulip brand hooks. Stay far away from Susan Bates hooks. They’re terrible for lace, which is too bad because they’re so easily available.
Boye brand hooks I use for lace are size 10-14. Tulip brand hooks have a different sizing, in that brand I use size 14-23. Just like with the string, the higher the number, the smaller the hook. Tulip hooks are nice because they have a golden coating on the tip that is smooth against the thread.
Most people who crochet are used to yarn, and have never used hooks and string this fine. Not to worry, you can do it, it just takes practice to get the hang of the smaller stuff. When I began, I thought it would be pretty impossible, the hooks are so small! But I discovered that it isn’t that different, and now I actually prefer thread to yarn. Lace thread never splits!
Lacis sells tiny crochet hooks, also lace making threads (Cebelia (all sizes & colors), Cordonnet, tatting thread, Flora etc. They also have a book catalog section of their site for reprints of antique patterns and excellent imported ones.
Herrschners All DMC and Coats & Clark threads, including Opera. They sell Optima online, and other exotic threads in the Yarn Shoppe paper catalog. Boye crochet hooks.
Ebay — you can find almost anything here.