The bulletin will have it - or the calendar you get from your church.
We start with Tone 1 on Pascha and then cycle through all 8 tones.
For Byzantine chant, some use Byzantine notation - very squiggly looking - and some use Western notation in English. Seems to depend on the chanter and the parish.
Fr. Seraphim Dedes of the Greek Orthodox Archdiocese of America has been working for some time on putting Byzantine chant into English and western notation.
Just a note on regular Sundays:
There will be the hymns from the Octeochos (Book of the Eight Tones) for the Resurrection. You'll hear the troparion and Kkontakion of the week. Then there will be hymns for the saints commemorated that day, or a particular feast, if still in the time it's being celebrated (for instance, Ascension is on a Thursday, so on the Sunday after, you will hear the troparion and kontakion for Ascension in with the Resurrectional troparion and kontakion).
Note: I've been told the Slav liturgical tradition (OCA, Russian Orthodox Church, ROCOR, Serbians, Ukrainians) tend to use more kontakia at Divine Liturgy than the Greek liturgical tradition (Greek and Antiochian/Arab)
So, for this past Sunday in my OCA parish-
Seventh Sunday after Pascha - also the commemoration of the Holy Fathers of the First Ecumenical Council and the after feast of Ascension
Sung in this order at Divine Liturgy:
Resurrectional troparion Tone 6
Ascension troparion Tone 4
Holy Fathers troparion Tone 8
Holy Fathers kontakion Tone 8
Ascension kontakion Tone 6
Big NOTE! :D
You need to remember there are two types of melodies under each tone.
Sticherion Used at Vespers: "Lord, I call" - Aposticha; Matins: the Praises
Troparion: for the various troparia and kontakia
Sticheraric - Used at Vespers: "Lord, I call" - Aposticha; Matins: the Praises
Irmologic - for the various troparia and kontakia
The tones used at the Divine Liturgy will be the irmologic tones in Byzantine chant. This is important when looking up music online. You don't want "Lord, I call" because that's sticheraric and is totally different from the irmologic used for troparia and kontakia.
I was a Matins chanter for 5 years at my previous Antiochian parish (western notation) and even went to a 3.5-day Byzantine chant class. I'm now a 3.5 year member of my OCA parish choir (and sing at a lot of different OCA events locally), in a choir that does all the great Russian liturgical music, including Bortniansky, Chesnokov, Archangelsky, and Kedrov Sr., but all in English. And lots of my favorite Znamenny chant, which is the ancient Russian church chant, very similar to Byzantine chant in some aspects. When you listen to and sing Znamenny chant, you can see how it was derived from Byzantine chant, which came to Kiev when the 'Rus were converted to Christianity in 988.
And believe it or not, YouTube is a great source for Orthodox liturgical chant to listen to and learn!
Also, at the Divine Liturgy, many things are not in any particular tone - they're just sung or chanted.
Things in a specific tone:
Prokeimenon (the Psalm verses before the Epistle reading - this word actually means "that which comes before")
In the Russian tradition, the alleluia before the Gospel will also be in a specific tone, usually the tone of the week. Not sure about the Greek tradition.
But the rest of the Liturgy - the Cherubic Hymn, the Anaphora, the litany responses, will just be chanted, or sung. Although from looking at YouTube, it appears there are Cherubic Hymns in at least several of the tones.
Just for fun, to compare the two traditions, side by side:
Bortniansky's Cherubic Hymn No. 7 (one of my faves!):
Byzantine chant Cherubic Hymn
I'm a church music geek. :D I don't read music very well, but I'm at Vespers and Liturgy all the time (I rarely miss, unless I'm out of town or at a special event at a friend's parish), which has really helped.