In English we generally say "wearing" unless we are putting something on at that moment, like "wait a sec, I have to put on my jacket" as opposed to "I'm wearing my jacket to dinner tonight." But Duolingo hasn't taught "wearing" yet, so should I assume that in Spanish it is more common to say "putting on" than "wearing" when referring to something one isn't actually wearing already?
True BUT... personal objects as well as body parts usually use the definite article but "mean" the personal pronoun.
"Ella lleva la cartera a la tienda." Actually means "She takes her purse to the store."
"A ella le duele la pierna." Actually means "Her leg hurts." Or "Her leg is hurting her."
Today I put on the brown jacket. Is it wrong because it's past tense?
No, you're not the only one. :)
Although Spanish also has a progressive form for verbs, it prefers to use the simple present. Unlike English, which has tight restrictions on when to use the simple and progressive forms, using the Spanish progressive is completely optional and usually only done if the action is in progress at this moment and if that progress is somehow important.