This page is not aimed at priests and religious who have an obligation to recite the Office. It is aimed at the laity who want to acquire this good and beneficial habit.
Which Hours to recite
In a word: you are free to choose.
Planning your prayer life is like planning your exercise or your diet. It is easy to get carried away by enthusiasm and end up surrounded by prayer-books (or exercise equipment or diet books) that you never actually use.
So - start slowly. Work out how much time you can devote to praying the Hours, and halve it. That way, you are likely to build a solid lifetime habit... and you can always add to it later.
Choose a regular time each day for your use of the Liturgy. This may be the morning, before the cares of the day overwhelm you; or it may be the evening, after the day's work is over and you can give your undivided attention to the things of the spirit. Whatever time you choose, stick to it. Don't keep changing your mind about when you do things. A regular schedule is essential.
If you miss a day, don't try to catch up.  Forget that you missed yesterday - just carry on with today. Otherwise you'll find yourself feeling guiltier and guiltier about the things you've missed, and your backlog will get longer and longer. The Divine Office is about joy, not guilt!
If a thing is worth doing, it is worth doing badly. Of course it is better if you are able to give the proper amount of time to the Hours that you have chosen, but if you don't have enough time one day, it is better to recite that Hour in a hurry rather than to leave it out altogether. That way the good habit isn't broken. (Of course, if you regularly find yourself hurrying and not getting the proper benefit, you will have to look seriously at how you are using your time).
But which hours?
See which ones suit you best. The Office of Readings is a rich source of material for meditation, especially on the days when Universalis contains the text of the Second Reading. Morning and Evening Prayer are more prayerful.
The daytime Hours (Terce, Sext and None) are very short and you are sure to have time for them if you try.
Always try to have a look at About Today. This won't always contain much, but we are gradually filling it with short details of the saint of the day, meditations related to the readings. So when there is something, it's worth reading.
What is the Invitatory Psalm?
It opens your celebration of the Liturgy of the Hours, and it should be recited before the first Hour that you say in the day.