Learn German: Ich nehme eine Jacke und eine Tasche.

In this sentence, eine Jacke and eine Tasche are indeed in the accusative case because they are the direct objects (the things being taken). However, for feminine nouns, the accusative article is exactly the same as the nominative (subject) article: eine. Only masculine nouns change their indefinite article in the accusative (becoming einen).

The English translation says "I am taking". Why isn't there a word for "am" in the German sentence?

German doesn't have a present continuous tense like English does (forms using "am", "is", or "are" plus an "-ing" verb). The single German present tense verb nehme handles both meanings. Depending on the context, ich nehme translates to either "I take" or "I am taking".

How do I know to use eine instead of ein for Jacke and Tasche if I haven't memorized their genders yet?

Both Jacke (jacket) and Tasche (bag) end in the letter -e. While there are exceptions, the vast majority of inanimate German nouns ending in -e are feminine. That is a fantastic shortcut to help you guess that you need the feminine article eine.

Do I pronounce the "h" in nehme?