Besaid Village : After the scene in the temple, you’ll get to see what the heck an Aeon is. After, you also get to give it a name (by default, this Aeon is Valefor). At night, speak with Yuna and you’ll get a question from Wakka, to which you can respond as desired. Speak with Wakka once more thereafter and you can finally get on with things.
To begin, Wakka will hand you the Brotherhood , a sword once used by his brother. It currently just has the Strength +5% ability (which makes it already better than the Longsword), but there’s more to come with the sword later on (including Strength +10%!). Anyhow, after the gifting, head into the item shop. Speaking with the girl with the dog eventually yields something along the lines that "her dog found something". Once that has happened, go into the third of the eastern houses of Besaid and speak with the dog to get a new Overdrive for Valefor, Energy Blast - he’s the only one to get TWO Overdrives, so be happy! (You’ll also see it in FFX-2, which can surprise some of you later.)
Don’t forget to stock up in the item shop! The main thing you’ll want in the coming hours are Antidotes for Kilika, and maybe some Phoenix Downs - we’ve got Potions out the wazoo here, not to mention Yuna’s Cure, so you ought to be fine on the HP front. Leave town and, as you do, you’ll gain the Map .
Village Slope : As you leave Besaid, you’ll be engaged in a battle against some enemies, and you’ll be taught how to utilize your party members for their roles. Basically, it works like this - Tidus for fast monsters (wolves), Lulu for magically-weak or elementally-weak enemies (Flan), Wakka for flying enemies, and Yuna to summon to wreck s***. Or to heal. Yeah, that’s helpful, too.
The first battle emphasizes Tidus and Wakka. In the second, you’ll use Lulu to hit weaknesses - Fire hurts Ice and Ice hurts Fire, while Water hurts Lightning and Lightning hurts Water. (Which means to use Thunder on the Water Flan.) Further east, there will be another scene.
Promontory : Pray if you wish here before moving on. Save as you go along to the Ancient Road because you’re about to face another boss.
BOSS - Kimahri
Strategically speaking, you won’t see a lot out of this fight. The boss in question will stick to a basic physical which hits for about 75 damage around now, while his Jump - occuring every three or four turns - deals closer to 125 damage. Even so, you can’t do much about it. You mostly just get to hit back-'n'-forth, healing as needed, since it’s just Tidus fighting here. Buffing yourself with Cheer might help, though. Eventually, a scene will finish the fight.
After the battle, Kimahri will join the party. Continue on. (Be sure to revisit the shop in Besaid, since it now sells equipment! Don’t bother with a Seeker’s Ring, though, since you’ll get a superior one momentarily!)
Waterfall Way : As you go along the linear path here, you’ll end up walking into a fight with a Garuda. This first fight will make you have Yuna call upon her Aeon. You’ll also be able to use Wakka’s special abilities to deal with another Garuda further ahead ( Dark Attack can blind the target, lowering their accuracy dramatically). Continue along the path to the crossroads, then southwest to the beach.
Beach/Dock : As you reach this area for your departure towards Kilika Island, speak with the people wishing you good-bye: they’ll also hand you items, including 400 Gil, a Remedy, an Ether, a Seeker’s Ring (with HP +10%), and three Phoenix Downs . Continue aboard the ship.