Adjectives modify objects.

Adverbs modify verbs.

When you say something tastes good, you are talking about the SOMETHING that is GOOD.

When you say "I am well", well is modifying how you are, if that makes sense. 

P.S. It is perfectly acceptable to say "I'm good/I'm feeling good"

So to clear it up, adjectives describe things, and adverbs change the way verbs act.

You can use a word in two ways to differentiate:

1. He is faster - Here, faster is an adjective, because it describes HIM, and not how he IS.

2. He runs faster - Here, it's an adverb, because it describes the verb, how he RUNS.

My school teaches adverbs like this:

verb+adjective=adverb.

Review:

A verb is the action such as "I am going" or "I went".  Verbs tell you what is happening in the sentence (or what the subject has done to the object). An adjective is used to describe the noun such as "The car is red."  In this case, red is the adjective because it is telling what color the car is.

Adverbs:

For example, a very basic sentence is "He is going."  "He is going" does not tell you how the action is happening.  This is why we have adverbs.  Basically, you use an adverb to add more description to the action.  A sentence with an adverb would be "He is running."  As you can see, the word "going" was replaced by the word "running".  "Running" describes the action, because now you know how he is going.

Note: Adverbs are not as important in conversation as they are in writing/reading (at least from my experience).

I hope this helped.  If it didn't, feel free to message me and ask me more questions.