It warms the cockles of your heart is a saying about good wine.  It comes from the Latin cochlea cordis meaning the ventricles of the heart.  (Brewer:  Dictionary of Phrase and Fable)  Nothing to do with cockle shells, then!

Heart-warming is often used about a comforting story.

My heart missed a beat.
This expression is about the effects of shock or fear.

Home is where the heart is.
A popular saying which could be about homesickness or the ability to settle where your loved ones are.

Eat your heart out means to worry unduly or obsessively.

From the bottom of my heart means sincerely.  The heart is regarded as the seat of the affections.

Heartfelt means sympathetic.

She took it to heart.
She was very upset.

Plucking at your heart strings is a way someone tries to gain your sympathy and perhaps motivate you to act.

Stick your chest out.
This instruction to stand up straight also means be proud.

The heart is mentioned many times in the Bible.

Man looks at the outward appearance, but the Lord looks at the heart.
1 Samuel 16:7