There are two types of saves in Majora's Mask.
Saving at an owl statue is a temporary save. It will save your progress, exactly as it is, at that point in time. You can save at an owl statue and then turn off the game, and when you come back, you'll be at that owl statue with all of your current progress. However, if you then turn off the game without saving at an owl statue, you'll revert back to the last time you saved by traveling back in time. In other words, always save at an owl statue before turning off the game!
Now the progress you save by traveling back in time is always permanent, regardless of whether you use owl statues. However, when you travel back in time, all events reset (including progress on dungeons), and all ammunition (arrows, bombs, rupees) are lost. You can save rupees by dropping them off with the bank in West Clock Town before traveling back in time. YOU WILL KEEP ALL MASKS AND KEY DUNGEON ITEMS. This is how progress is made -- you dedicate each three-day cycle to getting whatever items and masks you can acquire.
So play the game like this:
- At the start of a three-day cycle, SLOW DOWN TIME by playing the Song of Time backwards. This will make each in-game hour (real-life minute) last twice as long. (The Scarecrow in West Clock Town teaches you this.)
- Next, try to reach a dungeon. Usually to get to a dungeon, you will have to do a sequence of tasks or jobs in order to solve a problem, culminating in you learning a dungeon song from someone. There are 4 such dungeon songs in the game. Using them in a particular spot will give you access to the dungeon. When you get to the dungeon, you'll find a new owl statue. Activate it so you can warp to it later.
- Go back to Clock Town. Drop off your rupees at the bank. Reset time, saving all of your progress.
- Now it's a new three-day cycle. Slow down time again by playing the Song of Time backwards. Go out into Termina Field to stock up on arrows and bombs. And then use the Song of Soaring (taught to you by the owl in Woodfall) to warp to the owl statue at the next dungeon you'll attempt. (NOTE: You don't have to do any of the pre-dungeon tasks! Because you know the song from the previous cycle, you can skip all of the pre-dungeon tasks and start immediately on the dungeon.)
- With several real-life hours to complete it, you now have more than enough time to complete the dungeon. Even if you only get halfway through, and acquire the key item, it'll be quicker working through it on the next three-day cycle, since you keep that item.
- Beat the dungeon, return to Clock Town, deposit your rupees, and restart the three-day cycle to permanently save your progress and give yourself another 72 hours.
- IF AT ANY POINT YOU WANT TO SAVE AND QUIT, find an Owl Statue and save. Always save at an owl statue. Saving at an Owl Statue lets you pick up right where you left off in the current cycle; in contrast, playing the Song of Time does save, but it resets the cycle and you lose all your rupees and consumable items.