I've been learning scales and I love to play them in this fashion.
I can't seem to find a description anywhere of "proper" fingerings where there isn't a bunch of overlap. Proper is contained in quotes because I'm confident that there are multiple approaches to this. I'm fairly new to piano, so I apologize if this is a silly question. I'd just like to be pointed in the right direction.
On the ascending part for right hand, common approach is to use the following fingers: 1 - 3 | 2 - 4 | 3 - 5 (using | to separate the thirds) and repeat (1 - 3 | ...), so you place the thumb next to the pinky every 6 notes. Depending on the score, it might seem practical to start with an additional 1 - 2 before actually starting this progression, this would be: 1 - 2 | 1 - 3 | 2 - 4 | 3 - 5 and repeat from 1 - 3 (don't use this approach if practicing this particular score with both hands).
It's similar for the left hand: 5 - 3 | 4 - 2 | 3 - 1 and repeat, although the "practical" start doesn't apply in this case as the distance and mobility from the pinky to the ring finger is way less than the thumb to the index.
The descending part has the same principles, for right hand use 5 - 3 | 4 - 2 | 3 - 1 and repeat. For left hand would be 1 - 3 | 2 - 4 | 3 - 5 and repeat. For left hand the 1 - 2 might apply at the start.
This example will work smoothly for notes at C major which are all whites. If you play thirds on different scales or major at a key other than C, it's very likely that this progression won't work quite as well and will have to redefine the fingering.
There is no definite guide as you already mention, people have different finger lengths and hand sizes and some approaches might seem more comfortable for some and uncomfortable for others, so try experimenting and finding the best way for you.
As already said, there are a number of possible plausible fingerings and you should decide what's best for you.
But I, personally, would recommend 1-3 | 1-3 | 1-3... for the right hand, and 3-1 for the left. It's straightforward, and you can make better use of your hand's back and fourth movement without barely moving your fingers.
If black notes come into play, avoid finger 1 on black keys, by using 2-4 in these cases. So, in C minor, for example, with Eb, Ab and Bb, I would use:
Right hand: 1-3 | 1-3 | 2-4 | 1-3 | 1-3 | 2-4 | 2-4...
Left hand : 4-2 | 3-1 | 3-1 | 4-2 | 4-2 | 3-1 | 3-1...
Unlike EzLo, I don't think 3-5 is a good idea if you are planning on playing this fast, because 4-3 and 5-1 transitions do not favor that back and fourth hand movement I talked about, that is really helpful in faster tempos.
Welcome! There just cannot be a 'one size fits all' with fingering. One of the fun bits of practising anything on piano is working out your best fingering. Obviously this can be best done by you - with the help of your teacher, if you have one.
Ezlo gives some plausible fingerings, which need trying out, but there can't be a best answer for this. There might be a best answer for you, when someone happens to come up with what works for you, but I'll pretty well guarantee that someone's personal favourite won't suit everyone!
Next problem will come when you practise the same idea in other keys. Basic idea: keep the thumb for the white keys. Not always possible, so fiddle about and find a fingering that works for you.That will be the best answer to the question!
