Learn Russian: Мы покупаем только эту белую рубашку и это серое пальто. - We are buying only this white shirt and this grey coat.

The shirt is the direct object receiving the action of the verb покупаем (we are buying), so the phrase goes into the accusative case. For feminine words, the pronoun эта changes to эту, the adjective белая takes the -ую ending to become белую, and the noun рубашка changes to рубашку.

Why don't the words in это серое пальто change endings in the accusative case like the shirt did?

Because пальто (coat) is an inanimate neuter noun. In the accusative case, inanimate neuter nouns, as well as their matching adjectives like серое and pronouns like это, look exactly the same as they do in their nominative (dictionary) form.

I've seen это used to mean 'this is' with feminine nouns. Why is it used here with пальто?

The word это has two jobs. It can be a fixed word meaning 'this is' or 'it is' (like in это рубашка - this is a shirt). But here, it is a demonstrative pronoun meaning 'this' that must match the gender of the noun. Since пальто is neuter, it uses the neuter pronoun это.

Even though пальто is neuter inanimate, shouldn't it have some kind of case ending since it is the direct object?