Hi there.
I just started to learn bass guitar. I have a question.
Coming from a guitar background, when I would begin a scale like say ABC on the top string, my index finger would be holding position on the A, as my other fingers stretch to the other threads.

I understand I have to practice stretching, but considering the distance between threads on the bass guitar, how important is it to move to other threads while keeping your index finger in position ?

Thank you.
It really isn't any different than playing the guitar in that regard.  The less your hand and fingers move, the faster and more efficient your playing will be.  When you go from a 25" scale guitar to a 34" scale bass, it seems like quite an increase in distance between the frets.  But once you get used to it, you will find it is not that big of an increase in practical terms.  If you have problems making a stretch, then you will have to re-position your hand.  So it is a question of what works best for you.
"Maybe this world is another planet's hell?" - Aldous Huxley
As someone who went from guitar to bass, another tip is that if your hands are small, make sure your thumb is behind the neck acting as a pivot, and don't choke the fretboard.  Actually Scott's bass lessons has a great video on this:

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I am practicing without amp for now. I hear a lot of buzz on the left hand since the fingers are not strong enough yet.
But is it a matter of pressing down very hard because of the thick strings ?

Not necessarily.  In fact, pressing down harder on the strings can make fret noise even worse.  As you know, the electric bass has much thicker strings than does the guitar, and you are probably using roundwound strings like the vast majority of bass players.  These "telephone cables" can cause noises that you just don't encounter on the guitar.  The buzzing could be because your action is too low, or you may have a high fret or two (a common problem these days, even with expensive instruments), of it could be related in some way to your fretting technique.  Without being there to see and hear you play, it is difficult to diagnose.

One of the most annoying problems associated with the electric bass is "fret slap."  This is that God-awful noise (sometimes even actual tones) that are cause when you fret the string, and it "slaps" against the higher frets.  Short of getting a fretless bass or raising your action to the point where you can barely play, there is little you can do about it.  It is a big reason why, once you start playing gigs or recording, a good noise gate pedal will be indispensable.
"Maybe this world is another planet's hell?" - Aldous Huxley
Ahh cool. Thank you. Makes a lot of sense, and definitely some of the things I've across so far.
I know this is early days and I have no intention of going pro. I just want to make sure I have a good foundation.

And thank you for your continued input.
Really appreciate it.
To gauge how much pressure you should apply when fretting, lightly touch the string just behind the fret. Keep playing with the plucking hand. At first, you won't make a note, just a dull thud at best. Start to apply pressure progressively until you finally get a CLEAN note. That is how much pressure you need to apply fretting. You'll be surprised how little it actually has to be. Practice your fretting touch, and you'll be able to move around the fretboard quicker.
Quote by Dechirure
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I am practicing without amp for now. I hear a lot of buzz on the left hand since the fingers are not strong enough yet.
But is it a matter of pressing down very hard because of the thick strings ?

Practicing bass without an amp is pretty difficult -- you should have a very light touch with the left hand. 
It's worth spending a few bucks on a cheap headphone amp: http://www.rondomusic.com/product4521.html
Thank you very much. I will spend time on that. I actually have a small amp. I just felt like I should get going on the fingers a little bit.
But I watched a tutorial by Billy Sheehan, and he also recommends playing with an amp early on in your training.
Thanks again.