Learn German: Wir kaufen neun Eier und zehn Tomaten. - We are buying nine eggs and ten tomatoes.

No, basic numbers like neun (nine) and zehn (ten) do not change their endings based on the gender or plural form of the noun. They stay exactly the same.

The translation says "We are buying", but where is the "are" in the German sentence?

German does not have a separate continuous tense like English does. The single phrase wir kaufen covers both "we buy" and "we are buying". You do not need to add a translation for "are".

Why doesn't German just add an "s" to make Eier and Tomaten plural?

German has several different ways to make words plural, so you can't just rely on adding an "s". The singular Ei (egg) adds -er to become Eier. The singular Tomate (tomato) adds -n to become Tomaten. It is a good habit to memorize a noun's plural form as soon as you learn the word itself.

How do I pronounce the eu in neun and the z in zehn?

The eu combination in neun sounds like the "oy" in the English word "boy". The letter z in zehn is always pronounced with a sharp "ts" sound, just like the end of the English word "cats".