The main options for teapots are glass, ceramic or porcelain. I think these materials are all fine from a taste and hygienic point of view. Tea pots made out of glass are usually not very decorative. I still like them because one can easily see when they are dirty and I can see the color of the tea.
The strainers are usually made of glass, stainless steel, or even gold mesh or plastic. I don’t like plastic strainers, but I guess nowadays some high-tech plastic may be equally good as the other options.
My first consideration is not the material, but the size. If I just savour one or two cups from a special blend, then the size of the pot needs to be small. If I drink tea for breakfast with my partner, then we use a pot that holds five to six cups. To be honest, normally we stand in front of our tea collection and each one of us chooses a different blend. Hence, we brew two small pots. If friends and family are over for tea, then we use the biggest, almost two-liter ceramic pot.
My second consideration is how well it pours tea without dripping. Shapes of spouts that do not drip are well known. Strangely, only very few teapots use these forms of spouts.
Cleaning a teapot is a delicate matter. Most tea fanciers do not use soap as a small residual of soap can ruin the taste of the next brew. To a much lesser degree, you have the same problem if you use a single teapot for all different brews. If you brew a delicate white or green tea after having used the teapot several times for string black tea, then most likely you compromise on the taste. Consequently, I have different teapots for different groups of tea.
There is an enormously rich culture of artistic teapots. Even if they don’t satisfy the above conditions for practicality, you could still enjoy displaying it on a shelf – next to the tea storage boxes which might not hold tea either.
I heard several times that experts use two teapots in the process: One to brew the tea, then decant it into a second teapot from which they pour it at the table. I don’t see much of an advantage except that in this case you can put the tea leaves straight into the first pot which leaves them a lot of space to develop the aroma.