Is there any kind of modal verbs (can/could, may/might, must, will/would, shall/should) in Finnish? And how do I use them in sentences?

See a translation

Yes there are.
Can = osata, pystyä, kyetä, taitaa, voida (Osata and taitaa have a connotation of possessing a learned ability to do something, whereas pystyä, kyetä, and voida are more about being able to do something more simple.)
May = saada (also similarly to English people use voida, although in Finnish it's not considered a mistake at least to my knowledge)
Must = pitää, täytyä, joutua (Pitää is more an internal "must", whereas täytyä and joutua express more often things that are forced upon you by others)
Will = aikoa
I might be forgetting some synonyms for these.

In a sentence you only conjugate these verbs, while the main verb is in infinitive. (Edit: While typing the examples, I just realised that pystyä and kyetä are exceptions, where the main verb is in some conjugation, I don't know the name to. :D)
Examples:
Minä osaan soittaa pianoa. = I can play the piano.
En pysty vetämään yhtään leukaa. = I can't do a single pull-up.
Voitteko olla hiljempaa? = Can you guys be more quiet?
Saanko tulla sisään? = May I come in?
Meidän pitää lähteä nyt kotiin. = We must go home now.
Sinun täytyy nähdä tämä! = You must/have to see this!
Joudun mennä huomenna kouluun = I have to go to school tomorrow.
He aikovat rikkoa viestijuoksun maailmanennätyksen = They will try to break the world record in relay race.

Then we have the could, should and would and that's complicated enough to warrant an entire blog post, which someone has luckily made. http://thefinnishteacher.weebly.com/konditionaa...

This is a really broad topic so I'm not confident that I have answered it conclusively so I hope somebody else can maybe say more on the subject.