When you are learning something new, always go through the following steps:
1) Clap and count the rhythm (you can't play anything properly until the rhythm is secure).
2) Say the note-names (don't write them all under the notes - learn to read the music).
3) Play slowly, counting as you play. Also try saying or singing the note-names as you play.
4) Check all the details (techniques, fingering, notes, dynamics, articulations, rubato etc…).*
5) Speed up only when you can play your piece slowly and correctly all the way through.
If your new piece is long, just practise one small section at a time.
If you are unsure about the steadiness of your counting, use a metronome. If the ticking confuses you while you play, clap and count with the metronome until you are completely secure with the rhythm. Then try playing with the metronome.
Backing (CD, online audio or midi):
This is a fun way to learn your pieces. You can choose a slow speed and do your clapping and counting / playing with the background music. However: never use your backing as a substitute for working out the rhythm and notes - don't get into a habit of merely "playing by ear".
Be clear about what to practise:
First check what we've discussed in lesson-time (posture, techniques, fingering, hand positions, rhythm, dynamics, rubato, articulation etc.) and decide specifically what to work on for each piece.
Defining your goals:
Ask yourself these two questions before starting practice: "What do I want to achieve here?", and, "How will I achieve it?", ....and this question when you have finished: "What have I achieved?".
Make sure you come away feeling you've achieved something, no matter how small.
When you make a mistake:
Don't go back to the beginning and play the whole thing over again! You'll just keep making the same mistakes over and over, and get frustrated. Go to the part that needs extra work and sort it out before playing through the whole piece, noting other bars that also need special attention.
Keep your daily goals small and manageable:
There are no shortcuts! Be patient and persistent. With music, we improve performance in small, steady, "baby-steps". It may feel slow at the time, but it leads to the fastest improvement.
*Dynamics = loud & softs. Articulation = staccato & legato. Rubato = changes of speed.