How to pray using the service books without a priest, while traveling or at home…

When Orthodox people have no opportunity to attend Orthodox divine services, especially in non-Orthodox countries, then the Church allows and encourages individuals and groups of Orthodox to read the service books privately, for the preservation of their faith. Such readings have long been customary in monastic establishments, hospitals, schools, on shipboard and, in recent times, by Orthodox in the USSR and in the diaspora. Reading prayer books or service books may, at least to some extent, replace church services.
Besides preserving our Orthodox faith, reading services is beneficial because:
1. It teaches us, even in non-Orthodox lands, to remember and honor Orthodox feasts and saints' days.
2. It acquaints us with the order of church services and with the profound content of our service books.
3. It safeguards us from the danger of sectarian and heterodox influence
4. It helps parents and teachers raise their children and young adults in the spirit of Orthodoxy.
5. It unites dispersed Orthodox people in our faith and love for the Orthodox Church.
Orthodox Divine Services
The daily ecclesiastical office consists of a cycle of services that covers the entire 24-hour period. Since the church day begins with the evening, the order of daily services is: 1) Vespers, 2) Small Compline, 3) Midnight Office, 4) Matins, 5) First Hour, 6) Third and Sixth Hours, 7) the Liturgy and 8) Ninth Hour. Orthodox laymen may read or chant some portion of all of these, except the Divine Liturgy, which is replaced by the Typica.
In addition, it is permissible to read canons and akathists, either separately or as part of another service.
A canon is a collection of hymns in nine odes that honors the Savior, the Mother of God, a saint, a holy day. or a spiritual theme.
An akathist is a song of praise in twelve parts that glorifies the Savior, the Mother of God, a saint.... An akathist may be read or sung, or read with the refrains sung. 
How Laymen Read Service Books

The reading of service books should be conducted according to the following rules:

1. All [reader's] services are to begin with the exclamation: "Through the prayers of our holy fathers, O Lord Jesus Christ our God, have mercy on us. Amen.

2. All the priest's prayers and exclamations are omitted.

3. In place of the Great and Augmented Ectenias and the Ectenia of Supplication, "Lord, have mercy" is said twelve times; in place of the Small Ectenia, three times.
4. The Gospel is not intoned, but read in an ordinary voice.
Note: Every Orthodox Christian is obliged to read the Gospel privately, according to the ecclesiastical lectionary found in church calendars.
5. All other hymns, psalms and prayers are read or sung as when a priest serves.
6. The Typica (in place of the liturgy) may be read as indicated in Appendix 1.

The Order of Services on Feast Days

Since laymen are often involved with work and may not have time to read services in the ordinary week days, we shall give directions only for the festal services. 
On weekdays, the daily morning and evening prayers could be combined with Small Compline and Midnight Office, as desired.
On feast days, it is important to devote more time to God and to observe the feast with the appropriate reading and hymns. On the eve of the feast one may read Vespers, Matins and the First Hour, in the place of the All-Night Vigil. In the morning, one may read the Midnight Office, the Third and Sixth Hours, if desired, and the Typica. The evening of the feast, one should read the Small Compline with the proper canon or akathist of the feast.
The order and content of the services depend on the free time available and on the service books at hand. Here are more detailed instructions for three kinds of feasts: 1) Sundays, 2) the Twelve Great Feasts and other holidays of the Lord and of the Mother of God, 3) saints' days, our name-saints or ones we especially venerate.
1) Sundays

On Saturday evenings we read Vespers, including the stichera and troparia according to the tone indicated in the calendar. In the morning (or on the eve), we read Matins and the First Hour. At Matins we may read the Resurrection canon for the appropriate tone, or, if not available, the Canon to Our Sweetest Lord Jesus (in the prayers of preparation for Holy Communion) may be substituted. 'Me stichera for the aposticha, the troparia and the theotokia are according to the tone of the Sunday.
If Vespers and Matins are unavailable, then on Saturday night one may read Small Compline with the Canon and Akathist to our Sweetest Lord Jesus.
On Sunday morning we should read: the Midnight Office for Sunday, with the morning prayers and the Typica (the order for Typica is given in Appendix I).
Finally, on Sunday evening. we may read Small Compline with a canon to the Mother of God (either to one of her wonder-working icons or any other available).
2) Feasts of the Lord and of the Theotokos

On these feasts, including all of the Twelve Great Feasts, it is customary to read the proper service from the Festal Menaion. Vespers and Matins according to the Vigil are read, while the stichera, troparia, etc., come from the Festal Menaion. The canon of Matins is to the Lord or to the Theotokos, depending on the feast.
If the Festal Menaion is unavailable, then one may read Vespers (or perhaps Small Compline) with the canon or corresponding akathist, and one may take the stichera from the General Menaion, using the "General Service for the Feasts of the Lord" or "of the Mother of God."
In the morning: the Third and Sixth Hours and the Typica, with the troparia and kontakia of the feast sung in the proper places.
In the evening: Small Compline with the Canon of Repentance to Our Lord Jesus Christ, or the Supplicatory Canon to the Most Holy Theotokos (Paraclesis).
3) Saints' days

If there is a service to the saint in the Festal Menaion, then Vespers, Matins and the First Hour are read as usual, with the stichera, troparia, etc., from the Menaion. If there is no service to the saint, then we read from the General Menaion, taking the stichera, etc., from the general service to the class of saint being commemorated: i.e., to a hierarch, to a monastic, to a martyr, etc. At the polyeleos or perhaps at the end of the service, we chant the megalynarion to the saint (see Appendix II). In the appropriate places we insert the name of the saint being commemorated.
If neither the Horologion nor the Menaion is available, then we may read Small Compline with the canon or akathist to the saint, if available. (A church dedicated to that saint might allow us to copy the proper canon or akathist, so that we might read it on a nameday or other feast days.)
In the morning, we read the Midnight Office, the Hours and the Typica, with the troparia and kontakia to the saint at the Hours, and the kontakia of the temple, and of the saint or the day of the week, at the Typica.
In the evening, we read the canon to the saint; but if there is none, then the canon for Saturday to all the saints.

An article by Archpriest Sergei Shukin

Source: http://orthodoxinfo.com/praxis/services_nopriest.aspx