There are a wide variety of quests in the game, and you will get them from a variety of sources.
Often a villager will want you to do something for them. Quests that come by someone knocking at your front door of a morning, cannot be refused, but have no time limit. Quests that come by mail, someone asking for something, often can be refused - but there is no need to; they have no time limit, and there is no sign of repercussions if you are slow to complete them. So you don’t have a keg to make Pam her Pale Ale? She’ll wait! So you’ve never met anything that drops Void Essence? Don’t worry - you’ll get there. (In fact, that’s a quest which will only arrive in your mail when you’re getting close to the right levels in the mine. Some quests are keyed to your achievements, not the season.)
The only quests with time limits are the ones posted outside Pierre’s shop. These can be completed the day you get the quest or the following day. They usually involve an item that you should have available in that season, or collecting something, killing something, or catching something and then reporting to the appropriate villager (kill quests involve reporting to Lewis, fishing quests are raised by Willy, mining quests mean Clint.)
You need to finish these quests by the end of the day after you find the notice.
Sometimes the difficult part is finding the villager! This is easier if you have reached 2 hearts of friendship with the villager as they may be spending time in their bedroom. There is at least one website which records observations of where each villager may be found at particular times, depending on the day of the week, season, weather, etc.
In order to position yourself to fulfil short and long term quests, try to collect a little bit of everything. You may not be very impressed with garlic as a crop - but plant a few anyway, and stow at least one in a chest. It can be useful to organise these by keeping one chest for each season to hold the items peculiar to that season. (And an “all season forage” chest, a mining chest, a minerals chest, … use the Colour option to help you keep track, and place chests near the equipment you’ll use them with.) Arguably the bundles in the Community Centre also constitute quests, and the effect is much the same - you want to keep one of pretty much everything for the Community Centre. But the rewards are quite different.
More Complex Quests
Some quests are a little different. When Robin writes that she lost her axe in the forests south of Marnie’s Ranch, she means the same general area where you found Spring Onions in Spring. You’ll need to cross the river.
The Mayor’s Purple Shorts are in someone’s bedroom. (They can in fact be spotted from outside the bedroom, barely.) You’ll need to make friends with that person (2 hearts) so that you get permission to enter the bedroom, after which you can collect the shorts and return them to the Mayor. Returning them Discreetly does NOT mean putting them in your Grange display in Autumn … This quest’s implicit backstory is subtle enough to pass over the head of most kids, and to no more than amuse those who do get it.
Linus’s Blackberry Basket is on a part of the map you might not realise exists. It’s not hard to find the bus - what’s not obvious is that even if the bus isn’t going along the road, you can walk along the road … this is not the only hidden map.
As a step in solving the Goblin Problem you will want a Void Chicken, and they’re interesting and profitable anyway. If you’ve found a lot of artifacts & minerals, so that you have Sewer access, Krobus can sell you an egg. Or you can invite random chance; if you have an upgraded coop and it isn’t quite full, a Witch will *sometimes* fly over and leave you a Void Egg.