The first thing that you should know is that
Fate/EXTRA is a PSP game and cannot be played on a PS3. I know lots of PSP games can be played on the PS3, but this one can't. When you first launch
Fate/EXTRA, you will be given the option to start a new game on Easy or Normal Mode. There is a warning that there are "multiple dead ends" so it is recommended that you create multiple game saves in case you run into one. I highly recommend that you listen to this warning. You can save pretty much anywhere except inside the arena. I do wish that you could save inside the arena as well, since it can get really long and it would be nice to be able to save, but that adds to the difficulty of the game.
Fate/EXTRA is not meant to be an easy game. If you need to pause inside the dungeon, you can just turn the PSP off and it'll pick right back up when you turn it back on.
In Fate/EXTRA, you are a student at Tsukumihara Academy. Ms. Fujimara is the teacher, and she's a bit clumsy. Shinji Matou is the most popular guy in school, unnaturally so. Sakura is Shinji's sister. Mr. Kuzuki is the new, very odd teacher who gives you the creeps. All of this seems like a normal school where you are on the school paper and you go home and sleep every night. But new students keep showing up and acting strangely. When a mysterious girl in red appears and tells you that you're an NPC, we know something is really not quite right. You'll notice that your days in school are counting down from 4. After you run out of days, you will find out what is going on. The school is just a construct. All the real people in it are skilled hackers and magi who are there to participate in the Holy Grail War. In this world, the magi (and you) do not actually do the physical fighting. Instead, you will choose a Servant to bond with. The class that you choose will affect your gameplay. You can also choose to be a male or a female player. Even after you find out that you are a magi, you still have no memories. Something went wrong with the restoration process, so you are at a bit of a disadvantage and have to learn everything over. SE.RA.PH is the artificial world that the school occupies. Basically everyone there is a master hacker just to get into it, not that you remember how to do this. All of the magus there have "digitized their souls" and inserted them into the reality. It's not a talent that can be taught. You have to be born with the natural skills.
The goal is to beat all the other Magi in the tournament, but you're going to have to get a bit stronger to do this. Each day you have to prepare, you will need to go to the arena and fight the opponents there so that you can level both Master and Servant up. When you leave the arena it will end the day, so don't go into it until you are done with that day. This game isn't about fighting though. To fight well here, you are going to have to learn about your opponent so you have to spend time talking to everyone every day and researching everything that you can. If you miss information in a day, it is gone forever as people don't repeat themselves. Just keep on fighting in the arena to level up and get stronger so that you can beat the other magi and win the tournament!