I'm curious I started magic with my mother yesterday and she doesn't like house rules only official rules. Can I discard a card from my hand and draw a card from my library?
Discard at will? No. You either discard when instructed by a game rule (more cards than hand size at the end of your turn), when instructed by an ability (such as Enhanced Awareness's), or to pay a cost (such as for Rummaging Goblin's ability). If you aren't instructed to discard, you can't. Similarly, you can't just draw a card to replace any card you discard.
You only draw one card during your draw step. At the end of your turn, if you have more than 7 cards in hand, you must discard until you have 7 cards in hand. These are standard rules that always apply unless a spell or ability commands you to do otherwise. When discarding you are not allowed to redraw to replace the card unless a spell or ability allows you to do so. Library of Leng, Anvil of Bogarden, Grafted Skullcap, Heightened Awareness are examples of spells that affect how you draw and discard.