I wish to improve my aural skills, in particular the ability to hear something and play it straight away on the piano (not necessarily in the same key though, relatively will do). From what I have read it seems solfege is a good way to go. So -
What are the differences between sol-fa, solfege and solfeggio?
Does solfege seem like a good fit for my goal?
Does anyone here use solfege in their studies?
What are the skills gained from solfege that have made you a better pianist/musician that you use on a regular basis?
Do you use movable or fixed do?
I don't think solfege is what you want
Solfege/solfeggio is used to practice rhythm
In other words it is just counting of increasingly complex figures where instead of saying 1 & 2 & 3 & 4 you say the name of the note with the lating notation:
Do oo FaMi Si ii Reee
If you want to improve your aural skill you need to learn the particular aural feeling of each interval so that you can either read a sheet and being able to know how it sounds away from the keyboard or listen something and be able to reproduce it at the keyboard or write the sheet yourself
Each interval has a unique sound which gives the same feeling when applied to any tonality. So if you practice the recognition of that particular "feeling" you will be able to tell intervals from just listening them
My suggestion is to practice 2 intervals a week so that at the end of the month you have memorized them all and have greatly improved your aural skill
It's important to be hyper-consistent with the aural training, so even when you absolutely have no time at all to practice you must at least devote 3-4 minutes to aural training and interval recognition
Some hints in learning the sound of each interval:
Tone & Semitone [Major Second and minor Second]
Observe this pairs of note
C - D
D - E
E - F
Now sing in succession the pairs of note, starting from the same intonation for all them

As demonstrated by the sheet
Each pair of note must learn from the same base of intonation
The intonation can start from whatever sound as long as the relation of one tone between them is maintaned
If you do this you will notice that singing the three pairs C-D D-E E-F you're induced to sing them all in the same manner as if where all "tones"
This "induced mistake" is necessary to make one aware of the strong difference between a tone and a semitone
So now instead starting from the same intonation point, sing the corrent intervals

What you have to do is practicing singing the three pairs of note C-D D-E E-F so that you know the first two are tones and the third one is a semitone, starting from each note of the cromatic scale as an intonation basis
Third minor
This is a tone + a semitone so it's better to sing it in scale and then as an interval

What you have to is practicing singing first the small scale then the interval alone starting from all notes of the cromatic scale as an intonation basis
Third Major
Is a tone + tone, so again the best way to memorize it is singing it as scale and then as an interval alone starting from each note of the cromatic scale as an intonation basis
C - D - E || C - E
Perfect FourThe best way to learn the Perfect Four is to play it at the piano and consider that there's here a Tonic - Dominant
The Tonic is/represents a point of rest or of arrival
The Dominand is/represents a point of tension, or unresolution ... just like a bow in tension.
So after hearing the sound of the Perfect Four C-F at the piano try to remember the "particular quality" of this sound by remembering the "tension" or "bow in tension" principle
Practice the intonation of the Perfect Four starting from each note of the cromatic scale
Let me know if you think this may help me you and you will go on with the Fifth, Sixth and Seventh intervals