Learn Italian: Quando corro, sento il mio cuore.

The verb sentire is very versatile and is used for both to hear and to feel (and even to smell!). In this sentence, you could actually translate sento as either I hear or I feel. Both make perfect sense when talking about your heart beating after a run.

Why is it il mio cuore instead of just mio cuore?

In Italian, you almost always need a definite article (like il, la, i, le) before a possessive word like mio. The main exception is when you are talking about singular family members (like mio padre for my father). Since a heart is not a family member, you must include the il here.

Since cuore ends in -e, how can I tell if it is masculine or feminine?

Nouns ending in -e can be either masculine or feminine, so you often just have to memorize their gender. However, the words right before it give you a huge clue in this sentence: both il and mio are masculine singular words, which tells you that cuore is also masculine.

Why aren't there any subject pronouns like io (I) before the verbs?