Harvest Moon 64 is full of little tweaks and glitches that any true fan can figure out. Considering the game writers couldn't even spell the company name correctly any of the three times it appears, this isn't too surprising. There are far too many glitches in the game and mistakes the programmers made to list them all here, so I'm just going to cover the big ones.
Helpful Glitches
Some glitches are very useful for completing the actions required to get the party picture. So useful that you have to wonder if they are easter eggs intentionally left in the game by the programmer rather than glitches.
Karen's Fetish
Most of the time, giving a gift to a person only raises his or her affection level once per day. If you talk to Karen while holding your dog, however, it's a different story. Talking to Karen repeatedly while holding your dog will continually raise her affection level and, eventually, her heart level. However this only works early game and will not work after your dog's affection rises above a certain level. If you raised your dog's affection too high and want to make this glitch work, hitting you dog a few times with a tool until Karen says "Doesn't seem to clever..." will make this glitch work again.
Jeff's Question
Once you have made Elli love you, if you talk to Jeff (the Bakery Master) at the bar at night, he will ask you "What do you think of Elli?" You can answer that you love her or that she's just a friend. If you say that you love her, he will become jealous, but he will tell you that he hopes you do right by Elli. This makes him lose affection. However, if you say she's just a friend, Jeff will gain a good amount of affection, even if he appears unhappy. Better yet, if you leave the bar and come back in, he will ask you over and over again, gaining or losing affection each time. You can easily use this to your advantage to get him to dislike you or like you very quickly!
Magical Music Box
During the game, if you dig in a certain spot ten times, you will unearth a music box that your character buried there as a kid. To find it's exact location, search the big tree at the far end of the farm for the "Treasure Map" that will lead you to the exact spot. After you dig it up, if you dig in the same spot the next day, you will uncover another one! If you have more than one in your inventory at a time, however, Rick will stop fixing them, so make sure to only dig one up if you don't currently have one, broken or fixed, in your inventory already.
Free Betting
When betting on the horse and dog race, medals usually cost 50G each. If you spend your money, but hit 
to back out of the screen instead of hitting the "OK" button, your bets will be placed, but you will keep all of your money.
Happy Cows
If you allow your cow to get sick, then heal it with potion bought from the Green Ranch, the cow's milk quality will automatically go up by one. This is not recommended if you are going for the Party Picture, though, as it requires you to allow livestock to fall ill.
Raining Glitch
In this game, it will never rain on a festival day. If the weather channel says that it will rain, and the next day is a festival, you will notice that it is still bright and sunny. However, crops will appear watered and animals will still be unhappy if left outside. In other words, the day contains all the characteristics of rain, without it raining at all!
On the other hand, if you harvest a vegetable on a rainy day, the plant will suddenly be unwatered even though it is actively raining.
Easter Eggs / Hidden Features
Full Moon Berry
This isn't a glitch so much as it is an easter egg. Any day between Fall 10th and 12th, the Full Moon Berry appears outside the restaurant on the mountain. If you hit "A" at the trees in front of the restaurant, you will pick up the yellow berry. It doesn't do much, nobody is particularly crazy about it, but sells for 500G.
Climb the Mountain
Before the bridge is built, it is impossible to get to the upper portion of Moon Mountain... or is it? If you hit "A" near the base of the tree near the cave, you will climb the tree and end up on the ledge on the other side of the bridge.
Harvest Sprite's Home
If you hit "A" near the small opening in the cave where the harvest sprite usually hangs out, you will enter the Harvest Sprites' den. There are two more harvest sprites there, but they all act the same. If you give a gift to one, it counts as a gift for all three, likewise if you speak to one, the other two will say the second dialogue.
The Weather Vane
While in the mine (only open in Winter) and you are lucky enough to make it down to the third floor, you may dig up an object that looks like a golden chicken. This is actually an Easter egg from the SNES game that was on the roof of Ann's shop. If you show it to Rick, he will comment that this weather vane belonged to his grandmother and take it from you. He won't even give you a reward for finding it.
Invisible Glitches
There are ways to turn practically anything invisible from your possessions and items to your animals and other people. There are even ways to make yourself invisible. Be careful, though, as, in some cases, invisibility is permanent. Make sure to save before you try any of these (unless you prefer it the new way) and don't save after.
Disappearing Dog
This glitch technically doesn't turn your dog invisible, but it will make him disappear forever. Pick up your dog and set him down in your bathroom. He will wander around for a while and eventually hop off into the darkness never to be seen again. He will still make noise, but he will never come back.
Invisible Horse
Ride your horse around your farm until 6:00am. At this time, you will be teleported back into the house. After snacking on your morning sushi, you will be riding your horse. When you jump off, it will become invisible. You can ride it around, but you can never take it outside and it will remain invisible.
Invisible Baby
When your baby is old enough to leave the crib and move around on its own, pick it up and walk to your bathroom. Set it down just outside the door. Pick it up, then open the door the bathroom quickly. If you do it right, you will emerge from the bathroom holding your baby, but he is invisible. If you press 
to identify what you're holding, it will say your baby's name, but nothing is there.
Invisible Main Character
Wait around your farm until it's almost morning. Just as the clock hits 6:00am, jump into the hot springs. If timed correctly, the game will reset the next morning while you are in the spring. After eating breakfast, you will switch to being wrapped in a towel, invisible (with a shadow), and normal randomly as you run around. If you try to get in bed to save the game, your character will be invisible under the blanket.
Continuity Glitches
These glitches seem to give your character superpowers. Do you want to walk through walls? Fly over trees? Well these glitches are for you.
Jump off the Farm
If you arrange two stumps diagonally on the edge of the farm (see the picture to the left), you can jump over them diagonally and off the page. You'll land in the black, free to explore the dark expanse surrounding the farm.
Stomp the Trees
At the top of the ramp going to the vineyard at the crossroads outside your farm, instead of going to the right into the vineyard, walk up and to the left. You'll walk up and into the black region. Walk to the left and down over the trees or go up and into town.
Windwalker
When the bridge is complete, you can still cross the gap without the bridge. Walk into the left post of the downhill side of the bridge. If you run towards it and slightly left, you will eventually run out over the chasm.