Learn Farsi: همه‌ی میوه‌ها تازه است، ولی بعضی از آنها گران است. - All of the fruits are fresh, but some of them are expensive.

When saying "some of" followed by a pronoun like آنها (them), Farsi strongly prefers using از (from/of). While you can often use an Ezafe to connect بعضی to regular nouns, with pronouns, بعضی از is the natural choice.

In the first half, we use an Ezafe for همه‌ی میوه‌ها (all the fruits). Could we use از there too?

No, همه (all) almost always connects directly to the noun using an Ezafe (written as ی here because همه ends in a vowel). You cannot say همه از میوه‌ها. بعضی (some) is special because it gives you the option to use از.

Since "fruits" and "some of them" are plural, why does the sentence use the singular verb است (is) both times?

In Farsi, when a plural subject is inanimate (not a human or an animal), the verb is usually singular! Because fruits are inanimate, both میوه‌ها (fruits) and the pronoun آنها (them) take the singular verb است instead of the plural هستند.