The Church teaches that the primary hours of the Divine Offices, the "hinges" of the day, are Morning and Evening prayer (Lauds and Vespers). These two hours are the ones we should try to fit into our day. But my own feeling is that for the purpose of learning to pray the Office, and for becoming comfortable and personally attached to it, there's nothing like Night Prayer (Compline).
Night Prayer is on a 7-day repeating cycle. No matter what the liturgical season, there is no need to flip from psalter to propers--everything is there for each day, about 3 pages' worth per night. Night prayer is short and sweet--just one psalm or--on Saturday and Wednesday--two very short psalms. Say Night prayer every night for a week or two, and you will have acquired the rythmn and feel for liturgical prayer.
What makes Night Prayer special is the the "bed time" character of the psalms and prayers. It's as if God were tucking you in for the night, reassuring you that everything is going to be all right, little one, now go to sleep and don't be afraid--I'll be here if you need Me. For example:
I will lie down in peace and sleep comes at once, for you alone,Lord, make me dwell in safety.(psalm.4)
Into your hands I commend my spirit (psalm. 31)
Protect us, Lord, as we stay awake; watch over us as we sleep,that awake we may keep watch with Christ, and asleep, rest in his peace. (antiphon for canticle of Simeon)
Night holds no terror for me sleeping under God's wings. (amtiphon for psalm. 91)
I will bless the Lord who gives me counsel, who even at night directs my heart. (psalm 16)
Lord, we beg you to visit this house and banish from it all the deadly power of the enemy. May your holy angels dwell here and keep us in peace, and may your blessing be upon us always.
Then there is the fuller sense we should always look for in the liturgy. We aren't just praying about going to sleep, but about dying, and receiving His loving reassurance about that as well. Indeed, the psalms of Tuesday' and Friday's night prayer are of a more sorrowful type, meant to put us in mind of Gethsemane and give voice to our own sorrows or those of others. But that refrain of ultimate trust and abandonment to Mercy: "into your hands" ties the whole day together into a package that we can give to him and forget about. Christ is here. Tomorrow is another day.