Learn Russian: Я покупаю новые ботинки.

Russian doesn't have a separate continuous tense like English does. The single present-tense verb покупаю (from the imperfective verb покупать) can mean "I buy" as a general habit, or "I am buying" right now. The context tells you which one it is.

What case are новые ботинки in, and why don't their endings change?

They are in the accusative case because they are the direct object (the thing being bought). However, for inanimate (non-living) plural nouns and their adjectives, the accusative case looks exactly the same as the nominative (dictionary) case. So новые ботинки keeps its dictionary endings here.

I thought the Russian word for boots was сапоги. What is the difference?

Russian distinguishes between boot heights. Сапоги are tall boots that go up the calf or to the knee. Ботинки refers to shorter, ankle-high boots, especially ones with laces like hiking boots, winter boots, or work boots.

What is the singular form of ботинки?

The singular is ботинок (a masculine noun). Notice that the -о- in the last syllable disappears when making it plural. This is a common pattern in Russian called a fleeting vowel, where an о or е drops out when you add case endings to the word.