The English language is truly complicated at times. Too many options and too many exceptions to the rules! Rather than attempting to give you a few rules to which there will almost invariably be a few exceptions, I'll simply give you a number of ways to say the same thing, using the sentences you've provided.
The infinitive, to play, can function as a noun and as the subject of a sentence (technically, a predicate nominative), as in
- To play a good match this evening is what I hope to do.
With the insertion of a few more words, including the word that, you can make your second sentence correct as well, as in
- I believe that to play a good match this evening, I will need to bring my best game. [To bring one's best game is an idiomatic way of saying I intend to play my very best.]
Your third sentence is fine, as is. Moreover, the word believe could be substituted with the word think, with very little (if any) change in meaning. In other words, you can say either
- I hope I'll play a good match this evening,
or you can say
- I believe I'll play a good match this evening.
As for the difference in meaning between hope and believe, the words I believe have the ring of certainty to them, while the words I hope communicate less certainty than I believe.
Which of the following two sentences conveys more certainty?
- I believe I'll attend the party tonight,
- I hope I'll attend the part tonight.
If you say the first sentence conveys more certainty, you're correct.