Learn French: Je cherche une boîte à thé pour la théière. - I am looking for a tea caddy for the teapot.

This highlights the exact difference between a container's purpose and its contents. A boîte à thé is a container specifically designed to hold tea (a tea caddy or tea tin), regardless of whether it is currently full or empty. If you said boîte de thé, you would mean a 'box of tea', focusing on the actual quantity of tea inside.

I translated 'looking for' in my head, but where is the word for 'for' after cherche?

The French verb chercher inherently means 'to look for' or 'to seek'. Adding a preposition like pour directly after it is a very common trap for English speakers. You simply connect the verb directly to the object you are seeking: chercher une boîte.

The word théière obviously comes from thé, but how does that ending work?

The suffix -ière (or its masculine form -ier) is frequently added to a base noun to describe a container or tool associated with that noun. Here, thé (tea) becomes théière (teapot). Another common kitchen example is sucre (sugar), which becomes sucrier (sugar bowl).

English uses 'I am looking', which is the continuous tense. Why does French just use the simple present Je cherche?

French does not have a direct equivalent to the English '-ing' continuous tense. The standard present tense Je cherche translates to both 'I look' and 'I am looking'. While you could use the expression Je suis en train de chercher to emphasize that you are actively searching right now, the simple present is much more common and perfectly natural.