What do I do if I have zero cards left in my hand at the end of my turn? Do I wait until the end of my turn then draw the usual one card at the beginning of my next turn or do I draw a new seven card hand?
When you get to your draw step, you must draw a single card, unless some effect lets you draw more or less. If you attempt to draw a card when your library is empty, you lose the game.
Even if your hand is empty, the rules remain the same, and you still only draw one.
Here is how the comprehensive rules states it.
The draw step is the third step of the beginning phase.
504.1. First, the active player draws a card. This turn-based action doesn't use the stack.
504.2. Second, any abilities that trigger at the beginning of the draw step and any other abilities that have triggered go on the stack.
504.3. Third, the active player gets priority. Players may cast spells and activate abilities.
During casual games with my friends we draw 3 cards during our draw step instead of 1 because that's how my brother taught me to play Yu-Gi-Oh (we played that before I found mtg) and it sort of transferred over to MTG in my head all my friends thought it was better then only drawing 1 so we use that.
However if you play in an official game then that's a big no-no.