Learn Russian: Вообще я не люблю кофе.

In this context, вообще translates to "in general" or "overall." It is a very versatile and common word in spoken Russian. Depending on the sentence, it can also mean "at all," especially when placed right before a negated verb (for example, я вообще не пью кофе means "I do not drink coffee at all").

Why doesn't the word кофе change its ending here?

Кофе is an indeclinable borrowed word. It never changes its ending, no matter what grammatical case it is in. It is also an important exception to noun gender rules to keep in mind: although it ends in -е, it is a masculine noun, so adjectives describing it must be masculine (like чёрный кофе).

Since люблю is negated (не люблю), shouldn't the direct object take the genitive case?

Russian does often use the genitive case for the direct object of a negated verb. However, for concrete, everyday nouns, the regular accusative case remains very common. Furthermore, because кофе is indeclinable, its form is exactly the same whether it is in the accusative or the genitive case anyway!

How could I add "I like tea much more" to this sentence to practice this lesson's topic?

To say "much more," you use the intensifiers намного or гораздо paired with the comparative adverb больше. To follow up this sentence, you could say: Но я намного больше люблю чай (But I like tea much more) or Но я гораздо больше люблю чай.