Glory to Jesus Christ!
John, much of how much you can do depends on which and how many texts you have available.
the usual order for Saturday night (Sunday) vespers is the seven stichera in the Oktoechos (8 tones) and three for the saint that is commemorated that day at Psalm 140 with the dogmatika from the Oktoechos. There may be another festal stikhera that is taken at glory...before the ...now and ever and the Dogmatika if it is a major feastday that falls on Sunday.
Unless a major feast occurs on Sunday, the prokeimenon is always, "The Lord is King". Likewise the aposticha and troparia from the Oktoechos are taken.
For weekdays not of Poleleos rank at Psalm 140, there are three stikehera from the Oktoechos for the day and three of the saint. On Sunday, Monday, and Wednesday evenings the Dogmatika of the saint is taken. On Tuesdays and Thursdays a Stavrotheotokion from the Oktoechos is prescribed. On Friday evenings the Dogmatika from the Oktoechos for that day is prescribed.
For two saints at Psalm 140 three stikhera are taken for each saint and the three from the Oktoechos are optional.
The Prokeminon appointed for the day is taken.
At the Aposticha usually the three stikera for the day from the Oktoechos are taken, with the theotokion from the Oktoechos. At the Troparion take the troparion of the saint, glory both now and ever and the theotokion of the day. For two saints, tropar of first saint, glory, second troparion, both now, and theotokion of the day.
Please note this response is higly simplified, and taking the texts for Vespers depends on whether you have the texts or not. Some Vespers books, such as the Vespers book published in Uniontown by the Sisters of St. Basil the Great, have "common" stikhera for saints, such as general stikhera for martyrs, monks, nuns, etc.
While incomplete and with few instructions the Uniontown Vespers book allows the service to be taken if you don't have a complete 12-volume Menaia for the entire year which have the texts for each day and explain how they are taken. With the Uniontown Vespers and Matins books you will at least have commons, an abbreviated Oktoechos, and the Matins book has troparia for every day of the year.
If all you have is Byzantine Daily Worship, the only stikhera texts included are for Saturday nights of the Oktoechos and only a smattering for other feasts.
For major feast days, you will need a Festal Menaion for the texts. For Lent you will need a Triodion and a Pentecostarion for the period between Matins of Pascha and Sunday of All Saints for the Vespers propers.
Hope that helps, my response is very abbreviated. For private recitation for someone just starting out in growing into the worship of the Byzantine tradition, I recommend starting with Small Compline if you have BDW or another Horologion. The Canons for Compline are beautiful and often neglected. After you are comfortable with that and the Small Hours you can work into Vespers and finally Matins.
I am definitely not trying to discourage anyone from praying Vespers, but just so you know Vespers really is meant as a corporate worship service, as you can see from the antiphonal nature of the stikhera. Sometimes these seemingly complex services are not the best place to start our own home liturgical practices and the building up our interior prayer life.
For readers Vespers without a priest, a few differences are taken in the structure of Vespers. The doxologies taken by the priest are never taken by a layman when praying the office. This includes "Blessed is our God" at the beginning of Vespers, the doxologies at the conclusion of the Our Father, the prayer for the bowing of the head, etc. Instead, "through the prayers of our holy fathers, O Lord Jesus Christ have mercy on me (us)" is taken by the reader.
For a readers Vespers, instead of the litanies, Lord have mercy 12X is taken in the place of the long litanies and 3X for short litanies, concluding with "Through the prayers..."
I didn't even get into the Kathisma or divisions of the Psalter. These revolve throughout the year but the first Kathisma is almost always taken on Saturday night, in the Slavic usages in the form of the abbreviated "Blessed is the Man" and following verses. For usual weeknights, especially if you don't have a Psalter, these are often omitted.
If you want a more detailed course Deacon Thomas Stadnik of the Russian Catholic Church and I team teach a class for the Stamford diaconal program called "Troparion of the Day ? A Practical Guide to Finding the Propers".
