My question is I saw some battles with them using the plot like Summoning Season, Building Season etc. They search and REVEAL THE CARD AT THE SAME TIME. Some battles do otherwise like the owner of the plot reveal the card chosen card first.

How is the proper ruling on this? Thanks guys.

If an effect instructs multiple players to do something and doesn't specify an order, the controller of the player must do so first (then going clockwise around the table).

If you revealed summoning season, your opponent can fetch a character after seeing what your choice is.

Keep in mind, however, that the natural "flow" of play often sees both players searching through their decks at the same time and revealing the card they choose as they find it, especially if their choice of card is not going to be much influenced by their opponent's choice. While this isn't technically the way to do it, it's not really a misplay - especially in a casual game. The rule stated by -Istaril is something that tends to be observed much more closely in tournaments than in casual play or pick-up games.

I always thought that it would be better otherwise (showing at the same time) because of the phrasing on the card. When revealed, choose an oppponent, search, reveal, shuffle. Also for the fairness and also because the plot cards flip simultaneously might as well to these kind of search effects. Thanks for the clarifications..