When a skill has no specialties the CRB rules are clear on how to handle a character gaining skills during their terms of service, and after finally mustering out through training. But, all the examples are for skills that lack specializations, like Vacc Suit.
First: A Traveller has no level in a skill with specializations (say, Science). They then during a term they get a level in Science (this could be just listed as "Science" or "Science 1"). The rules require that the level is put into one of the specialties (not a problem). For discussion sake lets say they take "Science (Economics) 1". Would they then also have "Science 0" or continue with no levels in Science beyond the one level in the one in economics)?
Second: A Traveller "Science (Economics) 1". They then during a term they get a level in Science ( listed as "Science 1"). Since the Traveller already has a level in Science, can they take a first level in another Science specialty, say "Science (Physics) 1" or do they just loose the level?
SKILLS AND TRAINING
Each career has four or five types of skill table associated with it - Personal Development, Service Skills, Assignment Skills (divided into three sub-tables corresponding to assignments), Officer and Advanced Education. In each term you spend in a career, pick one of these tables and roll 1D to see which skill you increase. You may only roll on the Advanced Education or Commissioned tables if you have the listed requirement (usually EDU 8 or higher, or a certain rank).
Skills can be listed with or without an associated level. If no specific level is listed, then you gain that skill at level 1 if you do not have it already, or increase its level by +1 if you are already trained in that field. If a level is listed, then you gain the skill at that level so long as it is higher than your current level in the skill. For example, Gambler 0 would mean you receive the Gambler skill at level 0. This is of benefit to you only if you have no Gambler skill.
Vacc Suit would mean you increase your Vacc Suit skill by +1 or gain it at level 1 if you have no Vacc Suit skill to begin with. If you have Vacc Suit 0, it increases to 1, if you have Vacc Suit 2 it rises to 3 and so forth. Streetwise 1 would mean you get the Streetwise skill at level 1. If your Streetwise skill is already 1 or more, then this is of no benefit to you. If you have no Streetwise skill, or it is only at 0, you jump straight to Streetwise 1.
Some skills have specialties - specialized, more focused forms of that skill. A Traveller picks a specialty when they gain level 1 in a skill that has two or more specialties.
For example, a Traveller might have Engineer 0, allowing them to make Engineer skill checks without the unskilled penalty. They might then gain a level in Engineer, at which point they would need to choose a specialty. For Engineer, this is a choice between j-drive, m-drive, power plant and life support.
If they choose j-drive, they would record this on their character sheet as Engineer (j-drive) 1. They would then make all Engineer checks involving jump drives at DM+1 but would still make all other Engineer checks at DM+0 (in other words, with no penalty). A Traveller can have multiple specialties in a skill - they might have Engineer (j-drive) 1 and Engineer (power plant) 2. They would therefore make checks related to jump drives with DM+1, checks related to power plants with DM+2 and all other Engineer checks with DM+0.