There are different methods of prayer; vocal/liturgical, meditative and contemplative. Specific prayers generally fall into one of these categories. Most rosary prayers are intended to be meditative. However, most of us are used to vocal and liturgical prayers (spontaneous, read or repetitive).
People who try to pray the rosary prayer as if it is merely a vocal prayer will often find it difficult, boring, distracting and unfulfilling. Meditative prayer requires some specific practices, and if we don't know or don't engage in those practices, then we aren't praying properly and will find the prayer unattractive. But there aren't many teachers pointing this out. Most instruction booklets for the rosary, including our own, merely present the specific words to pray without giving much guidance regarding how one should be praying.
Because everyone's brains are not "wired" the same way, one set of rules may not work for everyone. But following are the specific practices that have helped us pray the rosary prayer meditatively:
Pick a quiet place without interruptions: Meditative prayer requires more focussed attention than vocal prayer. A single external interruption can torpedo meditation. You may even need to pick a time early in the morning or late in the evening when others are unlikely to interrupt you.
Meditative prayer includes vocal prayer: In the case of the rosary, you should be audibly (if quietly) actually saying the words of the prayers. Do not merely "say them in your mind." Move your lips! The reason for this is that human beings are capable of speaking repetitive words 'on autopilot'; with a little practice you can do so without your mind needing to pay close attention to the words spoken. However, if you are not speaking audibly, but are speaking the words 'in your mind,' then the mind is occupied by the practice, making it more difficult for the mind to be freed to open itself to and to focus upon the actual mystery.
Meditative prayer is more than vocal prayer: Once you reach the point of the rosary that includes the mysteries, as your lips are speaking the words of the prayer, your mind should be focussing on an image of the mystery upon which you are meditating. Close your eyes to avoid visual distraction; alternatively, if you have some icons or related images before you, you might focus your gaze upon them.
It is possible for us to repetitively speak a prayer while our minds engage an image. For some people it is even possible to speak a prayer while the mind engages an image and mental concepts related to the mystery (studies have suggested that women are more inherently capable of this than men).
Don't worry about speed; though we are often pressed for time, it is more important to pray properly than to pray quickly. Apart from the method, itself, one of the significant differences between vocal and meditative prayer is that meditative prayer focusses more upon listening than speaking. The mental focus upon the mystery may, over time, open one's mind and heart to God in such a way that new insights into the spiritual life are revealed.
If you are unfamiliar with the specific meditations or prayers, don't feel bad about having to turn to a pamphlet or instruction page between decades. That's life. In time you will learn the order of the mysteries and not need to use a help card any longer.
Finally, it can take some minutes to enter into meditative prayer. We almost need a 'warm up' period. Theoretically, the introductory prayers to the rosary assist in this immersion. But don't feel bad about yourself or your prayer if you find the initial minutes of prayer more difficult. You might also take a couple minutes prior to beginning prayer to clear your mind of distractions and stress. Sit peacefully in a quiet place, in a comfortable position. Breath deeply. Relax your mind and body. Most importantly, ask Christ, himself, to assist you in your prayer.