Mistake

I'm taking a trip to Los Angeles on next Wednesday with my girlfriend.

"on" is not wrong, just unecessary.

Today I made a plan of places we should visit.

And I noticed that there was an error with my flight.

It was for a different day than the flight my girlfriend has.

I would have to pay $400 extra if I changed my flight.

The big day is in 5 days.

Today I planed to out some places that we should go and see.

And I noticed that there was something wrong with my flight.

Usually, you will need a connector to pull together thoughts in English. In this instance you jump from sight seeing plans to an error with your plane ticket. It is not coherent. This would make more sense:
"Today I planed out some places that we should go and see. [While I was looking over my plans] I noticed that there was something wrong with my flight."

It was the My plane ticket had a different day to departure day for Los Angeles from than hers.

I would have to pay more $400 more if I changed my flight.

It was too expensive for me to change my ticket so that I decided to stay at a hotel alone for one day.

I'm using "for one day" as an example. Feel free to edit it with the correct time difference between your flight and hers.

The big day is after in 5 days.

I'm taking a trip to Los Angeles (on) next Wednesday with my girlfriend.

"on" isn't necessary, but if you use it "on Wednesday" sounds more natural.

Today I planned to some places that we should go and see. Also,"Today I made plans for which sightseeing places we are going to go to during our time in LA"

And I noticed that there was something wrong with my flight.

It was a different day to depart for Los Angeles from her.

I would have to pay more $400 more if I changed my flight.

$400 more, $400 extra. <--words come after numbers in these kind of situations.

It was expensive for me so that I decided to stay at a hotel alone.