as in, I know where the trainers are, However, in training, what does going up in a weapon skill actually do in the game mechanics. Mastery, High Mastery and Grand Master? I know it is different than in Baldur's Gate or Icewind Dale 2e rules. my THAC0 did not change when going up is there any speed factor adjustments as in Balder's Gate II or other perk? If not, what?

I usually played mage in PS:T but iirc weapon specialization worked similar to BG and IWD

Your base thac0 doesn't change but if you use a weapon of the type you put proficiency points into you get bonuses to hit and damage. On top of that you can attack more often per round for reaching specialization and grand mastery with it.
If your game is unmodded (Qwinn changes this) you should get the same benefits as in Baldur's Gate and Icewind Dale, my character did.
Don't bother about speed factor, it's useless anyway.

so in short, you only get SF adjustments I am more interested in knowing the non-modded version I have tried downloading mods on sites but always seem to aquire malware from unmanaged locations.

just wanted to know the breakdown of Planescape Torment(1)'s
specialization
mastry
HM
and GM but it seems like what you say it is similar to BG but not the same. I am going to assume by your long explamation that it is like Baldur's Gate 2 rather tha 1 or IWD 1 that means it goes likek this
Spclst= +1 to hit +2 damage
Mastery= +1 to hit(additoinally)
HM = a speed factor(of some variety)
GM = the +1 damage bonus addition to what you get in Specialist
if that is the case, not too bad but not as good as BG 1 still if I had a safer link to a mod that fixed this without needing to download a "mod manager" I'd try it. :)

It's more like in IWD:

* no penalty
** +1 to hit +2 to damage +1/2 attack per round
*** +3 to hit +3 to damage +1/2 attack per round
**** +3 to hit +4 to damage +1/2 attack per round
***** +3 to hit +5 to damage +3/2 attacks per round