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.
You can also use 'wish' with a noun to 'offer good wishes'.
Notice that when you want to offer good wishes using a verb, you must use 'hope ' and not 'wish'.
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.