While it's hip to talk about feelings in our Oprah society, we're not all good at it. A common trick is share a thought masquerading as a feeling. Starting a sentence with "I feel...." doesn't mean you're experiencing or communicating your feelings. I see people do this all the time. Take this couple:
Spouse 1: "I feel that you broke the garbage disposal."
Spouse 2: "I feel like you didn't look at it closely because you don't care."
Spouse 1: "Well, I'm just telling you how I feel."
Not feelings. In fact, any time "I feel..." is followed by "that" or "you" or "like", we're not talking about emotions. Most of the time, those are thoughts, beliefs, or judgements. No, you don't feel that NASA should pursue a mission to Mars. You believe it. It's your opinion. It's a thought, not a feeling.
Feelings can be broken down to some combination or degree of sad, mad, glad or afraid (some experts might also throw in shock, love and/or envy). That's it. Degrees of sad might range from bummed out to devastated, while anger might range from annoyed to enraged. Sometimes feelings combine, creating frustration (sad + mad), bittersweet (sad + glad), etc.
And what makes these feelings, rather than thoughts? Feelings can be felt, actually physically experienced. It might take a little practice, but a simple way to get in touch with real feelings is to become aware of the physical sensations you have in your body. When people experience emotion, their body tends to feel a certain way, unique for everyone. When people are sad they may feel a weight in the chest and shoulders. People feeling angry might report an expansive, explosive sensation in their chest and arms. Fear tends to be a weight in the stomach. Joy is often a light, energized feeling throughout the body. But again, it can be different for anybody.