While Zozo appears to be a normal town, it's more like a sprawling dungeon. There's a lot of Treasures to collect, Monsters to fight, and even puzzles to solve. Your primary goal is to reach the top of the tower with the Relics sign, but there are two other buildings worth exploring as well.
Zozo Encounters
Before exploring, you should be prepared for the dangerous Encounters you'll find in Zozo. There are two sets of Encounters, one for the outdoor area of town and another shared by all indoor areas. The outdoor Monster you'll likely get sick of first is the HadesGigas, a monstrously huge foe with 1,200 HP. Their normal attack is powerful, and their Head Butt attack deals quadruple damage. They can only use it every three turns, but it might be their opening attack. To make matters worse, 1/3 of the time when they die they'll use Magnitude8 to deal heavy damage to the whole party. The HadesGigas isn't vulnerable to much you can really exploit. They do take extra damage from Poison but Edgar is still better off using the Drill than the Bio Blaster. The one exception is Celes's Imp Spell, which will work and prevent Magnitude8. It's effective but does burn through MP. If you choose to attack normally, hit hard and make sure you keep up everyone's HP in preparation for Magnitude8. Or just run.
The best case scenario outside is running into a group of Gabbldegaks, weak monsters that offer little threat. They can cast Vanish every other turn, which can get annoying since you have to use a magic attack to hit them at all. The Bio Blaster is effective here as they have low HP and a vulnerability to Poison. Sometimes a pair of Gabbldegaks will appear with a Harvester. These are annoying enemies that will often use Potions to heal, and who will throw MithrilKnives at you if left alive long enough. They'll even counter a successful Steal by Stealing right back from you! Kill them quickly to avoid the devastating Throw attack, and focus on them first if you see one in a rare Encounter alongside a HadesGigas.
Most indoor encounters here will feature at least one SlamDancer. In a group, they aren't a significant threat, contributing modest attacks and sometimes inflicting Psyche with Daze Dance. However, when left alone they will use random tier-2 attack Spells, which is bad news for you. The simple solution is to kill them first in mixed groups, but you may encounter a single SlamDancer who will immediately start the spellcasting barrage. (If Celes happens to go quickly, Runic can be a good counter strategy.) The good news is that they have less than 400 HP so you should be able to defeat them with one or two good hits. You can sometimes steal a ThiefKnife from one, though it is not advisable to try when they are alone. (There's actually quite a lot worth stealing here—HadesGigantes have Atlas Armlets and if you're lucky you can nab a pair of DragoonBoots from Harvesters.)
Optional: Searching for Treasure
The first building you'll see in Zozo is the Cafe. This is a totally optional building to explore, but you'll find a Tincture as well as your first clock hint inside. The Tincture can be obtained by climbing to the top floor and taking the right door. This will lead to a balcony with stairs up to another floor. The door on the right contains the Tincture, while the one on the left is rusted shut and will not be opened for a very long time. Take note of what the man in the Cafe says ("Clock's second hand's pointin' at 30."). This is the first in a series of hints about the current time, which will be used to set a clock to obtain the Chain Saw later on.
To continue searching for treasure, head north from the Cafe. A man outside the Inn tells you not to jump between buildings, which is a hint for later. Inside the Inn you'll find a clock. Correctly setting the time will cause a path to the Chain Saw to open. Note the options you have for setting the clock, as they will help narrow down the solution from the hints you're given.
It may not be obvious, but you can walk west from here behind the tall tower. You'll find a Tonic in the Armor Shop. There are no Treasures in the Weapon Shop, but you'll find another clock hint. The man behind the counter insists that the clock in the building has no minute hand and is always wrong. The clock's one hand is pointing at the two, and you can therefore conclude that that's a minute hand and it's correct. The clock's minute setting should be 10. Once you're done exploring the side areas, you may want to leave town and rest up before tackling the long tower climb ahead of you.
The Main Tower
When you're ready, head southwest from the Cafe to the Relic Shop.You'll weave in and out of the building several times as you climb. Make sure to speak to the thieves along the way for various hints. The thieves in line will tell you it's every potential hour except 6:00, which solves another part of the clock riddle. When you arrive outside after following the line, you'll notice the hook of a crane nearby. You can use this crane to return to the ground floor, avoiding the line of thieves, if you need to head back and rest. There's a doorway several floors above the crane leading to the precious Thief Glove, but there doesn't appear to be a path forward. Remember what the man outside the Inn told you, and head back down to the floor with the open windows on the west side. You can walk out the window and jump across, making your way all the way to the West Tower.
This building can seem a bit confusing, but there are two western paths you can take to find some stairs leading up. The northern path will allow you to continue on. The pots at the north end of this hallway contain a Potion and a Tincture. The thief here and the one outside each give you hints about the clock's seconds. You'll jump back to the original tower and climb several more floors of uninteresting stairs. Once back outside, make sure to check out the door to find the powerful Fire Knuckle for Sabin. Continue on up to find Dadaluma, the boss of this area.
Dadaluma
The fight with Dadaluma is heavily scripted, and you can take advantage of that. He will call two Iron Fists to help after enough time has passed, and there's not much you can do about that except kill him before it happens. But he'll also counter Fight, Magic, and Steal in interesting ways. Steal is met with Steal, which should be no problem since you'll get back the GP he stole at the end of the fight. It's worth it for a shot at a Sneak Ring. He'll respond to two uses of Magic by Throwing a Dirk or MithrilKnife, then using Jump. He'll do the same Throw counter twice if you use Fight four times, though there isn't much reason for most characters to use Fight at all. Finally, when he's lost 1,350 HP, he'll use a Potion or Tonic three times (randomly chosen each time) and cast Safe on himself.
Dadaluma's basic attack sequence is much simpler. He'll usually attack normally for decent damage, but he can also Sweep to inflict Seizure in addition to the damage or use Shock Wave for moderate magic damage. He poses little threat outside of his scripts. If you really want to ruin his day, have Celes cast Imp on Dadaluma. She can also use Runic to counter Safe if you have good timing.
Terra and Ramuh
Once past Dadaluma, you'll enter a large room. There are two treasure chests on the sides of the room, one containing RunningShoes and the other an X-Potion. The main event is straight ahead: a transformed Terra, lying in bed. Ramuh will appear and explain the situation. He'll give you your next target: the MagiTek Research Facility. He'll also introduce you to the concept of Magicite, and you'll receive your first four espers: Ramuh, Kirin, Siren, and Stray. You'll find the party members you left behind waiting at the entrance to this room, and you'll be transported out of Zozo. You'll have a chance to reform your party, though you must include both Locke and Celes. You'll need to head all the way back to Narshe to change up your party again, so choose carefully.
The Chain Saw
A new tutorial ghost NPC appears near the entrance of Zozo in case you want a refresher on Magicite and Espers. You should now have all the hints required to go get the Chain Saw. You know from these hints that the hour is 6:00 and the minutes are 10, but it's left unclear whether the seconds are 10 or 50. (I suspect this is due to a problem with the dialogue, since two of the hints are redundant and one likely should have solved this ambiguity.) The correct answer is 6:10:50, and once you enter that, the path to the Chain Saw will open. This Tool is similar to the Drill, though more powerful. However, 1/4 of the time it will attempt an instant Wound attack. This is great when it works, but it deals no damage if it fails, making the Chain Saw a bit more of a gamble than the Drill.