Let's start off with the easy part. ' I wish to' can mean the same as 'I want to' but it is much, much more formal and much, much less common.
I wish to make a complaint.1
I wish to see the manager.2
You can also use 'wish' with a noun to 'offer good wishes'.
I wish you all the best in your new job.1
We wish you a merry Christmas.2
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Notice that when you want to offer good wishes using a verb, you must use 'hope ' and not 'wish'.
We wish you the best of luck.1
We hope you have the best of luck.2
I wish you a safe and pleasant journey.3
I hope you have a safe and pleasant journey.4
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However, the main use of 'wish' is to say that we would like things to be different from
what they are, that we have regrets about the present situation.
I wish I was rich.1
He wishes he lived in Paris.2
They wish they'd chosen a different leader.3
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