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.
I wish to see the manager.

You can also use ‘wish’ with a noun to ‘offer good wishes’.

I wish you all the best in your new job.
We wish you a merry Christmas.

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.
We hope you have the best of luck.
I wish you a safe and pleasant journey.
I hope you have a safe and pleasant journey.

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.
He wishes he lived in Paris.
They wish they’d chosen a different leader.

Notice that the verb tense which follows ‘I wish’ is ‘more in the past’ than the tense corresponding to its meaning.

I’m too fat. I wish I was thin.
I never get invited to parties. I wish I got invited to parties.
It’s raining. I wish it wasn’t raining.
I went to see the latest Star Wars film. I wish I hadn’t gone.
I’ve eaten too much. I wish I hadn’t eaten so much.
I’m going to visit her later. I wish I wasn’t going to visit her later.

In the case of ‘will’ , where ‘will’ means ‘show willingness’ we use ‘would’.

He won’t help me. I wish he would help me.
You’re making too much noise. I wish you would be quiet.
You keep interrupting me. I wish you wouldn’t do that.

Where ‘will’ means a future event, we cannot use ‘wish’ and must use ‘hope’.

There’s a strike tomorrow. I hope some buses will still be running.
I hope everything will be fine in your new job.

In more formal English, we use the subjunctive form ‘were’ and not ‘was’ after ‘wish’.

I wish I were taller.
I wish it were Saturday today.
I wish he were here.