This page explains how to complete the Sun Stone sidequest in Chrono Trigger.
Read The Fated Hour for instructions to get a Ruby Armor, which is helpful in this side-quest.
Go to the Sun Palace
First you have to find the Sun Palace, and the easiest way to do that is to fly over Porre in either time period, then travel to 2300 A.D., and there will be the island. Land and go in.
The next boss uses fire attacks, so equip things like Red Mail, Red Vest, and Ruby Armor/Taban Vest. This boss only uses magic attacks, and the characters with the best magic defenses are Marle, Magus, and Lucca. Have a healer in your party, and Magus can be the one wearing the Ruby Armor, so he will be getting hurt when others are healed).
Walk up to the sparkle to start the fight.
Son of Sun
You can Charm a Black Mail from Son of Sun (core), so if you want to steal it, bring Ayla, equipped with her Charm Top.
Do not attack the core, because it will counterattack and take no damage. Your target instead is one of the five flames surrounding the core. There are four "wrong" flames and one correct one. If you have been grinding, Magus will know Black Hole, which you can use to permanently remove some of the incorrect flames. If you hit the correct flame, the core gets hurt. If you hit a wrong one, Son of Sun will counterattack and take no damage. Once you've found the correct flame, keep attacking with physical attacks until the boss does a Roulette Shuffle. This will randomize the correct flame, so you will have to search for it again.
After you defeat Son of Sun, follow it. You have to go up some stairs to reach the platform. When you approach it, it turns into the Moon Stone. Check on it to get it.
In 65 million B.C., press select to see a small island at the upper right of the map. Fly and land there. Go in and check the sparkle to leave the Moon Stone.
Go to the same spot in 600 A.D. and check inside. There is a Tab to the left of the stone.
Next, check inside the Sun Keep in 1000 A.D. The stone is not there.
Then go to Porre in 1000 A.D. You will see some suspicious sparkles over the Mayor's Manor. Go to the Snail Stop in Porre, which is east of the Mayor's Manor, and buy some Jerky from the guy behind the counter.
Now go back to Porre, 600 A.D., the Elder's House, and give (DO NOT SELL) the Jerky to the woman in the kitchen.
Go to the Mayor's Manor in 1000 A.D. and talk to the now-generous Mayor, who will give you the not-yet-fully-charged Moon Stone. Fly to the Sun Keep in this era, and check on the spot to leave the stone there.
Put Lucca in your party and go to Sun Keep 2300 A.D. Get the fully-charged Sun Stone.
Afterward, you will get the WonderShot and Sun Shades.