There's no wrong way to play, and the first season you'll probably not be very efficient with what you do. There's nothing important that you can mess up. This game always gives you a second, third, fourth chance.
There are quests in the mail sometimes that seem overly ambitious. The ones in the mail don't go away, though, so if it takes you five years to figure out - for example - how to bring Pam the drink she wants, it doesn't matter - she'll still be happy.
The quests you can pick up on the board outside Pierre's shop do have a time limit. However, if you don't manage to fill the request in time, nothing happens and nobody gets upset. They just quietly disappear from your journal.
The best way to make money is by fishing. The other ways in the first weeks are farming and foraging. The training rod helps. You can eat stuff you forage or harvest, or you can sell it.
Your menu had tabs for friendship, for crafting, and for game options (among other things). I always find it helpful to scroll down and select "show tool hit location" in the game options because that really helps when you're learning to use the tools.
Other than that, it's fun to just go out and explore. If you are really stuck and get really frustrated, you can always ask and people will help you. Just try to be specific, because nobody wants to accidentlly spoil things for you...