The basic rule: the last one wins
There can only be one weather at a time, and the one that arrives later replaces the one before. The abilities that set it on entry (Pelipper's Drizzle, Tyranitar's Sand Stream, Torkoal's or Ninetales' Drought) activate in Speed order, from fastest to slowest. That is why, when two of them start the battle, the weather that stays belongs to the slower one.
An example: with the same Speed investment, Tyranitar (61 base Speed) is slower than Pelipper (65), so if they lead against each other, sand wipes out the rain. If your weather setter is faster than the opponent's and they lead together, you start in the opponent's weather; if it is slower, in yours.
And a detail people forget: setting the same weather again does not refresh it. If Pelipper comes back in while rain is still up, the turn count carries on where it was.
The exception: Charizard Y's sun
Charizard enters the battle without Drought: it gets the ability when it Mega Evolves, and Mega Evolution happens at the start of the turn, after switches and before any attack. So on the turn Charizard Mega Evolves, its sun wins: it does not matter that Pelipper or Tyranitar set their weather on entry, or even that the opponent switches Pelipper in that same turn.
Since Speed is recalculated during the turn in Champions, the change kicks in right away: Venusaur already moves with Chlorophyll that same turn and the opposing Basculegion loses Swift Swim before it attacks. The only way to beat it that turn is to Mega Evolve a weather setter too, like Mega Tyranitar: Mega Evolutions go in Speed order, so the weather that stays belongs to the slower one, almost always Tyranitar.
Five turns, and why a Mega never reaches eight
Weather lasts five turns. If the Pokémon that sets it holds the matching rock (Heat Rock for sun, Damp Rock for rain, Smooth Rock for sand and Icy Rock for snow), it lasts eight. But a Mega holds its Mega Stone and cannot hold a rock: Mega Charizard Y's sun and Mega Tyranitar's sand always last five turns. Pelipper, Torkoal or Hippowdon can extend theirs.
Count the turns carefully. If weather is set in the middle of a turn, that turn already counts as the first. If it is set when replacing a fainted Pokémon at the end of a turn, it lasts five full turns ahead.
What does not exist in Champions
Champions has no Utility Umbrella and no Safety Goggles. Nobody can ignore sun or rain with an item, and only Rock, Ground and Steel types and a few abilities avoid sand: everyone else loses a sixteenth of their max HP at the end of each turn. On top of that, Rock-type Pokémon get 50% more Special Defense in sand.
There is also snow, from Mega Abomasnow and Alolan Ninetales (Snow Warning). It deals no damage, but it raises the Defense of Ice-type Pokémon by 50%, and like any weather, it wipes out the opponent's sun, rain or sand.
How to use it in a game
At team preview, check who sets weather on each side and compare Speeds: if your setter is faster than the opponent's, it may be better not to lead with it and keep it in the back to set your weather again later.
During the game, keep track of the opponent's weather turns and use Protect or switches to wait it out. And if you bring Charizard Y, pick your Mega Evolution turn well: it is the only moment your sun is guaranteed to win.
On the matchups page you can see how sun, rain and sand teams do depending on what the opponent brings.
