The canonical hours (Agpeya) can be prayed at home alone, or as a family. The Psalmody (Vespers Praise, Midnight Praise, and Morning Doxology, aka tasbeha) can as well. The cycle of readings can be followed at home. A set of hymns in veneration of the saint of the day (aka tamageed) can also be prayed at home. The only parts of the liturgical day that cannot be private, and must be in the church as a community with a priest are the raising of evening and morning incense, and the Liturgy of the Word and the Liturgy of the faithful.
Recordings of the agpeya can be found at agpeya.org , and of the Sunday Annual Psalmody can be found at http://www.stantonymonastery.org/index.php?option=com_content&view=article&id=85:english-midnight-praise&catid=26:audio&Itemid=3
Before discussing what's done in the home, it's useful to have a brief outline of the full liturgical cycle of a day that can potentially be prayed in a church/monastery:
The general order of a non-fasting day is as follows:
-If prayed in the church, the 9th, 11th, and 12th hours are prayed together in the evening
-The Vespers Praise is prayed
-The Raising of Evening Incense is Prayed
-In the middle of the night, (or to make it easier, immediately after the evening Incense, or immediately before morning services) the Midnight hour of the Agpeya is prayed.
-The Midnight Praise is prayed
-In the morning the Prime hour is prayed from the Agpeya
-The Doxology of Prime is Prayed
-The Raising of Morning Incense is Prayed
-The 3rd and 6th hours of the Agpeya are prayed together
-The Liturgy of the Word, and the Liturgy of the Eucharist is prayed.
On days of fasting, the Liturgy is prayed later in the day to allow people to fast (i.e. consume nothing), since after receiving Communion no one may fast (i.e. must break the fast with vegan food). As such the prayer of the 9th hour is moved to after the 6th hour, rather than with the evening hours to reflect the time of the Liturgy. On days of strict fasting, the Liturgy is later still, so the 11th and 12th hours are prayed with the rest, and it is illegal to raise the evening incense since that is when the Liturgy should be.
So, clearly there is enough there to fill an entire day, which is not what is intended for laity to do.
Generally, one takes a rule or prayer from their father in confession, and follows it in obedience, neither neglecting it, nor adding to it (though their maybe some flexibility to the rule as needed to accomodate wordily obligations)
A typical starting point for such a rule would be to pray Prime in the morning, and Compline (the 12th hour) in the evening. But rather than praying the whole thing, which can easily take 15-20 min if done at a prayerful pace, someone will be started out with only the introductory prayers, then after that rule is established one can ask their Father in confession for more, who will typically add a few of the Psalms, and later the Gospel and litanies, and so on, until a point appropriate for that person is reached. Typically praying 3-6 of the Psalms and most of the rest of the hour, which can be done in 5-10 min. Many people work up to praying all of Prime and Compline, with many also adding Sext (noon).
It is central to Orthodox spirituality to do such things with gentle progress, and not to make jumps. Satan will tempt people and tell them that they can do more, and their Father in confession doesn't realize what they're capable of. Then people take their rule, not from the preist, but from satan, and lose the blessing of obedience, and find themselves burning out and being inconsistent in prayer, sometimes doing much, sometimes not wanting to any. It is better to do the amount prescribed in obedience. The same applies to fasting. One cannot defeat self-will by doing what one wishes.
Here is a set of sermons on Orthodox spirituality that cover these things in detail: http://www.stmaryscopticorthodox.ca/media_sermons/spirituality/spirituality.html
Or in book form:http://www.stmaryscopticorthodox.ca/content/books/spirituality.pdf
Another great introduction to Orthodox spirituality is "The Way of the Ascetics" http://www.amazon.com/Way-Ascetics-Ancient-Tradition-Discipline/dp/0881410497
Some families will pray the Midnight Praise together occasionally. The Doxology of Prime is a great place to start with getting to know that, since it's tune is fairly simple, and it is short.
Many families read and discuss the Bible briefly before evening prayer together. Normally people read the Gospel and litanies in tune, with the rest said, and often with the Psalms prayed silently. It is nice though to take turns with everyone chanting a psalm, or to simply read a few of the psalms together. Best to keep things simply to start though. After that it is important to have time for prayer from the heart. Some families do this aloud, but most silently. It is a wonderful thing to stand together praying silently, and the fell united in prayer standing before God without speaking aloud and worrying over finding words in front of each other.
Having an icon corner is more of an Eastern Orthodox devotion, as far as I can tell, but many Copts have adopted it as a useful anchor to a family prayer life in the house. One thing to note though is that in the Coptic tradition, people never burn incense, only the priest does. It's nice to gradually gather a collection of icons of the feasts, and some meaningful saints and to put the icon of the season or day in a central position to be able to follow the seasons and high points of the calendar as a family, but that certainly isn't essential. Reading the synaxarion is another great way to keep in touch with the calendar, but it is not of great quality in terms of translation and editing.
When starting a rule of prayer, it is important to keep it personal. Praying together as a family is great of course, but in general experiences should not be shared with the world. What this means is that no one should know about your rule besides your father in confession. This avoids vainglory. It may seem like a strange detail at first, but if you go through some of the spirituality resources above they make clear why this is important.