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 2 Antidotes. Once you've picked these up, head towards the people on the shore.
After the scenes, follow Wakka (use the Save Sphere if you want to). You'll encounter some monsters in the water; more Pirahnas. There's 2 Antidotes in a chest that's easy to spot along the water path. Once you reach the end, you'll go through a few scenes, then you'll be in Besaid Village. Check around a bit; there's some Antidotes, 400 Gil, and some Potions in some chests in a secret path area to the left, 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. Head towards the back and you'll find a Save Sphere to use if you need to. Once you're done in here (thee'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.
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.
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.