Here are a few exercises I have given my piano students, both online and in-person, to help them become more proficient at the aural tests in graded exams (especially ABRSM), and also to give them the skills to learn to play by ear.

Singing scales and intervals.

1.  Start off easy. Play a C major scale (or any other major scale you like) at a pitch/register that suits your vocal range. Then sing the C major scale without playing it.
2. Next play the C (or the tonic of whatever scale you have played), and without helping yourself on the piano, try and sing a major 3rd above it. Work it out by singing the C, D, E in your head first (i.e. the first 3 notes of the scale).
3. Then play the C and try and sing a perfect 4th above it. Sing the notes in between the C and F in your head to try and work it out. Check you’ve got it right by playing the F afterwards.
4. Do the same with a perfect 5th, major 6th, major 7th, and 8ve.
5. Next, mix up intervals, so don’t do them in the order they appear in a scale. I.e. give yourself a starting note, and try and sing a perfect 5th above it. Work it out by singing the notes in between the lower and upper note in your head. Then choose a different starting note and try and sing a major 3rd above it. Etc.
6. Next, choose more difficult intervals, such as minor 3rds, 2nds, 6ths and 7ths; and maybe some augmented/diminished 5ths and 4ths.

The purpose of the above exercise is to start to get you to associate particular intervals with sounds in your head, so that eventually you can look at a melody on the sheet music, see the sequence of intervals, and hear them in your head. Eventually you get better at a) working out the intervals, and b) remembering what the intervals sound like.

This works the other way round as well. Eventually you get better at hearing a melody and recognising the intervals that form the melody, and knowing exactly what those intervals are on the piano, so that you can more easily work out the melody by ear.

Sight singing from a score.

1.  Look at the start of the melody of a piece you’re about to sight read, perhaps just the first phrase. Give yourself the starting note on the piano, and transpose it to an octave that you can sing. So if the melody is in treble clef, and you have a low voice, you may need to transpose it down an octave to suit your vocal range. 
2. Once you have found your starting note, analyse the intervals formed by the melody. E.g. the melody may start with a G, the next note may be up a major 2nd, the next note might be up a major second again, the next note might be down a major 3rd etc.
3. Using the same technique you used for singing random intervals, work out how each interval of the melody sounds, and slowly try and sing it at sight. 

Harmony.

To get better at playing chords by ear, start off with some very basic harmony ear training - recognising the difference between major and minor tonality.

1. Get the sound of major and minor tonalities into your head first by playing a major scale followed by a minor scale starting on the same note. Do the same for all keys (or all the keys that you know). Also do the same for arpeggios and broken chords. Particularly notice how the third degree of the scale differs between major and minor. E.g. in an A minor scale the third degree of the scale is a C, in A major it is C#. In arpeggios and broken chords, this flattened third degree of the scale is the only thing that distinguishes minor from major.
2. Next, listen to recordings of pieces of music and try and decide whether they are major or minor. This is even better if you can check your answers afterwards by looking at the score and working it out, or getting your teacher to guide you.

Recognise chords.

To start to recognise individual chords, you need to have knowledge of the theory behind the chords, and be able to identify them. To start to become more familiar with chords, so that eventually you can play them by ear, learn some basic and commonly used chord sequences. The most basic and commonly used chord sequence in all styles of music is: I IV V I (1, 4, 5, 1). (It helps to know the theory. see an explanation here.) 

In C major this chord sequence would consist of the following triads:
C major (chord 1)
F major (chord 4)
G major (chord 5)
C major (chord 1)

Try playing this chord sequence on the piano in every key. Also experiment with different chord inversions. As you become more familiar with the sounds of these chords and how they relate to each other, you will get better at recognising them when listening to pieces of music.

Once you have done this with basic chord sequences, you can start to do the same with other chord sequences. See this article for other commonly used chord progressions.

Recognise cadences.

Cadences are pairs or sequences of chords found at the end of sections of music. It helps to be very familiar with the sound and musical effect of all the cadences.

1. A perfect cadence (V, I) makes a piece of music sound finished, a bit like a full stop.
2. An imperfect cadence (I, V or IV, V) sounds like the end of a section, but that there is more to come, a bit like a comma.
3. A plagal cadence (IV, I) sounds very hymn like ("amen").
4. An interupted cadence (V, VI) sounds like it was heading for a perfect cadence, but then takes the listener by surprise. It has the effect of building up the tension by slightly delaying the resolution created by the final cadence.

Practise playing all of these basic cadences in every key. Also analyse the scores of pieces of music in order to identify these cadences and then play them. After identifying and playing these sorts of cadences enough times, you will start to recognise the sound of them when you hear them. Then you can start to work out exactly what chords are being used, just by listening.

Once you become more familiar with the sounds of these basic cadences, you can start to learn to recognise the sounds of the chords that lead up to them too, as there are specific chord sequences that are used for specific effects. Again, it helps to know the theory behind this. 

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