Final Fantasy X/Besaid Island
You'll regain consciousness in a warm, tropical ocean. Things might be looking up — and you get hit on the head by a Blitzball. Not much has changed, then…
 Meeting Wakka and the Besaid Aurochs
Before you swim towards the shore and the people there, head right to one of the small coves at the edge of the accessible water. The chest here contains the Moon Crest, but it'll be a long time before you can use it (this is simply saving time for later). The chest on the bottom left of this beach area, by the shacks, contains 2x Antidotes. Once you've picked these up, head towards the people on the shore.
After the scenes, talk to the members of Besaid Aurochs, they will give items (except Letty) (5 Potions total, 200 Gil and 2x Hi-Potions), then follow Wakka (use the Save Sphere if you want to). You'll encounter some monsters in the water; more Piranhas. There's 2x Antidotes in a chest that's easy to spot along the water path, and two other chests which are slightly harder to find, which contain a Phoenix Down and a Hi-Potion. Once you reach the end, you'll go through a few scenes. Follow the path downwards, endure yet another cutscene, then you'll be in Besaid Village.
Check around a bit; there's 2x Hi-Potion, 400 Gil, and 2 Potions in some chests in a secret path area to the left (between the Item shop and the Crusader Lodge), and further down another chest with some more stuff. Have a look in the houses as well if you want, otherwise head to the house-tent in the top left of the village (don't
go in the temple just yet). You'll be introduced to the Crusaders. Pick up the Al Bhed Primer II, and then head towards the back and you'll find a Save Sphere to use if you need to. Once you're done in here (there's also a help console in here if you need to use it, for example if you missed or forgot the Sphere Grid tutorial), head outside and towards the large temple.
Introduce yourself to the summoner guy standing by the stairs, and check the side rooms if you want. Talk around — there's a guy who will basically outline everything you've been told so far. Head outside again, and go inside Wakka's house. If you missed anything, choose to leave; otherwise, accept Wakka's offer to rest. After Tidus' flashback, head out of the house and go back to the temple. Head inside, and talk with Wakka. You'll barge into the Cloister of Trials, much Tidus' regret.
 Besaid Temple: Cloister of Trials
This is the way the summoner and her guardians are tested — by a set of meaningless and supposedly dangerous puzzles involving glowing spheres. Anyway, examine the symbol on the wall you were facing with all the floating unreadable characters. A green glyph will appear on the right hand wall in front of the Yevon symbol on the floor. Go over and touch it, and the wall will rise through the roof, revealing some stairs. Head down, and interact with the wall slot on the way down to have a brief introduction to the types of spheres you'll encounter inside Cloisers of Trials. Glyph Spheres trigger all sorts of things, as do Besaid Spheres (Besaid Spheres get replaced by the respective temple name spheres, and do things loosely related to their temple). Destruction Spheres are necessary for getting the extras from each temple, and for doing some sidequests much later on in the game. For now, take the Glyph Sphere from the slot and take it down to the door at the bottom. Insert it and the door will open.
Ignore the slot on the outside wall around here for the mean time. Head around to the floating symbols on the next wall, and touch them to open up another section of wall leading to a room with lots of glowing blue lines. Take the Besaid Sphere from the wall and put it in the slot opposite the room you were just in. The wall will disappear. It is really important now that you don't push the pedestal that you get any further; if you push it onto the mark on the floor you won't be able to come back and get the Destruction Sphere treasure until right near the end of the game, by which time a certain absurdly difficult superboss will be waiting right in front of Besaid Village, barring your access. It's not worth the hassle of trying to get past it whilst really underlevelled, just to get the Destruction Sphere treasure (and a few other goodies that appear inside the village).
Instead of pushing that pedestal, go back to the door that you opened with the Glyph Sphere and put it in the spare slot you saw earlier (not the one in the room with the lines). The wall will rise, and you can retrieve the Destruction Sphere from inside. Take this and put it where you first found the Besaid Sphere. A purple line will go around to the apparent dead end around the corner, and blow a hole in the wall. Walk around and through the hole, and open the treasure chest. You've just got Besaid's Destruction Sphere treasure, the Rod of Wisdom.
Return to the pedestal. To access the final part of Besaid Temple, just push the pedestal with the Besaid Sphere in it in a straight line towards the Besaid symbol on the back wall. There's a square on the floor you need to push it on to, and the "shining glyph" that the help referred to is right next to the back wall with the flickering white light — if you shove a pedestal somewhere you don't want it to go, just step on the nearest one. Anyway, once you've pushed the pedestal onto the square, you'll be treated to another explanatory scene with Wakka, you'll descend to the Chamber of the Fayth. Here you'll meet the summoner's guardians.
After you get the Airship much later in the game, return to the Cloister and enter the Chamber of the Fayth after completing the trials get a cutscene and open some treasure chests containing: Potion x2, Elixir, White Magic Sphere, and a Hi-Potion.
 Meeting Yuna, the Summoner, and her Guardians
You'll leave the Cloister of Trials and gain control of Tidus again inside the main Temple. Head outside to the circle to watch the apprentice summoner perform her first summoning! Afterwards you'll get to name your new summon. Valefor is the default, so we'll stick with that — if you rename it (something femenine hopefully, since Valefor is a female), note that this guide will refer to it as Valefor.
Wakka will properly introduce you to the Besaid Aurochs. You'll also meet Yuna, the summoner. Ignore the old people, they just won't let you near her again for the mean time. There's nothing else to do, so talk to Wakka again and take up his offer to rest. When you wake up, you'll talk some more. After another rest, you can use the Save Sphere if you want, then head outside to meet Lulu and Yuna. You can now leave Besaid Village, but first go back into the Crusader's Lodge and look around a bit — you'll find Al Bhed Primer II.
If you wish to gain Valefor's second Overdrive, then go back to the Item shop, and speak to the shopkeeper. Afterwards, find her dog, and you'll receive Energy Blast. It'll be taught to Valefor automatically.
Now leave and head up the road. You'll run into some predetermined battles that you can't avoid. The first is against a Dingo and a Condor. As the in-battle dialogue suggests, each character has a type of enemy (at least during the start of the game) that they're useful against; Tidus is useful against wolves and fast enemies, and Wakka's more useful at hitting aerial targets. The next preset battle is against a Water Flan, where you'll be introduced to using Lulu and her black magic, specifically the elements. In case you don't get it, use Thunder on the Water Flan to kill it (it's amusing to use Water, Fire or Blizzard on purpose and see everyone's reactions).
You'll be back at the altar/statue thing at the top of the road. When prompted, make sure you pray — your party will like you more. Use the Save Sphere and then follow the red blinking arrow. Ignore the currently indecipherable Al Bhed on the pillar here (unless you used a compilation sphere) and the random Save Spheres that are out of reach. You'll be confronted by a random enemy with a rather large weapon.
 Boss: ??? (Kimahri)
You will have seen Kimahri before — in the Chamber of the Fayth and standing around outside the Temple just before Valefor's summoning. He's not particularly dangerous here — his attacks do 50-60 damage, but Jump does about 100-110. Heal if necessary, and use your overdrive, Spiral Cut, if you get the opportunity — it'll do about 350 damage and that's just a little under half his health.
Once you're done, you'll find out who exactly Kimahri is, and he'll walk off again. After some more walking, another preset battle will occur, against a rather large and menacing Garuda. Wakka doesn't want to fight it, so you have to switch Tidus out for Yuna, who then needs to summon Valefor. Once Valefor's out you can begin fighting it — a normal attack does about 200 to the Garuda, but you also have Sonic Wings at your disposal, which will do about half the damage of a normal attack but delay the Garuda's turns somewhat, making it more likely for you to get two turns in between the Garuda. Bear in mind that Sonic Wings won't always work enough to allow an extra turn, and some enemies/bosses are immune to delay. The Garuda has 1400 HP, which will go down pretty quickly. If you want you can wait until you get an Overdrive with Valefor, then use Energy Ray to do upwards of 1000 damage!
After the battle, carry on along the path. You'll run into another Garuda further along — but this time you can use Wakka's Dark Attack to inflict darkness on it. Now almost all of its attacks will miss. Darkness is another standard status effect as far as Final Fantasy games go, and it becomes a lot more common later on in the game. For this battle, attack with Wakka and Tidus (re-use Dark Attack once it wears off) and use any of Lulu's black magic spells to inflict about 250 damage. The Garuda won't last long, although you might want to bring Yuna in and either have her summon Valefor again or just use Cure on someone just so she participates and gets AP at the end. Ignore the random Save Sphere, and carry on along the path back to the beach.
Once you get near to the dock, make sure you talk to people — they all seem eager to give Yuna gifts which you can take on her behalf. The summoner dude from the temple will give you a Seeker's Ring, and the other people will give you an assortment of stuff, like an Ether, 400 Gil, a Remedy, and 3x Phoenix Downs. Although it's not absolutely necessary you'll want all the random stuff you can get. You're now on the S.S. Liki, heading to Kilika!