Stardew Valley has lots of quests to complete and this basic Stardew Valley quest guide should get you started.
Check your journal. Tasks will often be given to you seemingly out of nowhere and they all appear as a reminder here.
Check your journal
Remember to read your mail as this is another way you receive quests.
Read your mail
Progressing elements of the game will automatically trigger you to get certain quests, so engage in as many activities as you can find.
Tips
Some quests have a set time to them, so make sure you complete them before time runs out and you lose the opportunity forever.
When a quest is complete click the cash icon on your journal page to collect the cash and clear the task from your list of things to do.
What do you want to do now?