Learn Dutch: Ik draag een lichte jas. - I am wearing a light coat.

The base word for "light" is licht. In Dutch, when an adjective comes directly before a noun to describe it, you usually add an -e to the end. Because jas is a de word (de jas), the adjective before it gets an -e, even when we use een (a).

Would it still be lichte if I was talking about a "het" word, like a shirt (het hemd)?

No, and this is an important rule to remember! If you use een (a) before a het word, the adjective does not get the extra -e. So, "a light shirt" would be een licht hemd. But because jas is a de word, it gets the -e here: een lichte jas.

The translation says "I am wearing", but where is the "am" in the Dutch sentence?

Dutch doesn't usually translate the English "am ...-ing" form word-for-word. The single verb draag (from dragen, to wear) covers both "I wear" and "I am wearing". You don't need to add a separate word for "am".

Does lichte mean light in color, or light in weight?

Just like in English, the Dutch word licht can mean both! Without more context, een lichte jas could mean a pale jacket (light color) or a jacket that isn't heavy (lightweight).