General Farm Tips
The various "suns" you can cook are only useful for filling out your recipe book. Some people like them as gifts, but they're expensive gifts. You cannot ship them.
You can only have one horse, but if you don't have more than four hearts with it by the time it becomes an adult (a year after you get it) then Barley will take it away. He'll come back the next year with a new fully-grown horse for you, however, but it won't have any hearts.
Your son will never grow up - he always stays at a toddling stage. You can only have one son, and never a girl. If you want a game where your child can grow up, look at A Wonderful Life for the Gamecube.
You can control the weather by using your TV and saving your game. Save your game every night before you go to sleep. The next morning, check the weather when you wake up. If it doesn't look like you want it to, reload from the night before and sleep again. Eventually, you should get what you want. This also works if your animal is going to die - save the night before and reload if it dies the next day.
The animals on the mountain are just for decoration and cannot be captured. Other items that are solely for decoration include the greenhouse at the Poultry Farm and the boat at the dock.
Fences must be completely enclosed to prevent your animals from being scared by wild dogs. Even if a fence appears to be enclosed but has open corners, the dog will scare them though he cannot get in. Make sure the corners of the fence are closed too.
If you miss buying an item on the shopping network, don't worry. It will cycle around again later, but you might have to wait a while for it. Chances are, if you can afford the item at the time, you should buy it.
When you become friends with people they'll let you into their back rooms. They'll also say different things and some people will have events that generate based on your friendship level. Some people might give you recipes once you're friends with them, but to get the recipe stored in your book you'll need to remember it yourself and then go cook it first.
Mining Tips
The levels of the mine are randomly generated, so the stairs and stones are not always in the same place. It's possible for a level to load with no stairs down to the next level, so to avoid running into dead-ends and to conserve energy you should use both save files as described below. It's also possible for a level that is supposed to contain rare jewels (such as mythic stones, Alexandrite, or pink diamonds) to have as many as five of these rare gems on a level, or as few as none, so save the level above so you can go down and possibly find more.
To conserve energy and get to the really low levels of the mines, you'll need both save slots of your game. When you first enter the mine, save in slot 1. Then search for the stairs, find them, and reload slot 1. Go straight to where you found the stairs and dig them up in one hit, and save in slot 2. Then go down and save over slot 1. Then search for the stairs, find them, reload slot 1, dig up the stairs, and save in slot 2. Go down the stairs and save over slot 1 again. Repeat this process.
Mythic stones can only be found in the spring mine after you've upgraded all of your tools to the blessed level. They're found on the same levels as the Goddess jewels, which are 60, 102, 123, 152, 155, 171, 190, 202, and 222.
The teleport stone can be found at the bottom of the spring mine (level 255) beginning on Spring 1st of your 3rd year. You must have an open spot in your tools section to be able to find it.
Cursed tools are always found in the dirt in the winter mine by using your hoe, as are the stairs to lower levels. Gems, jewels, and stones are found in the rocks.
Girl Tips
To see a girl's blue heart event you must have the large rucksack from the supermarket and have an open spot in the items section. This is to receive the gift the girl will give you.
Items received from a girl's blue heart event are useless and can only be stored in your cabinet as a memento. You cannot give them back to the girl, sell them, or give them to anyone else.
Your wife will never leave or "divorce" you, ever, for any reason. It is possible to lower her heart level once you're married but she'll stick with you, even if she's unhappy. However, if you do not keep her at nine or ten hearts for several seasons after marriage, you will not have a child until you get her heart level back up for a while.
Rival events cannot be triggered if the girl involved has a higher affection level for you than she does for the rival involved. So if you're at a red heart with Mary, don't expect to be able to see any of her rival events. You can give her items she hates to lower her heart level a bit and be able to trigger the event.