Characters improve by training in a particular skill.
Mongoose Traveller Core Rulebook, Page 59:
LEARNING NEW SKILLS
The more skills a character possesses, the longer it takes him to
learn a new skill. A character’s Skill Total is calculated by summing
the levels of each skill (level zero skills count as zero). A character
with Mechanic 1 and Gun Combat (slug pistols) 2 would have a Skill
Total of 3.
To increase a skill, a character must train for a number of weeks
equal to his current Skill Total plus the desired level of the skill. So,
to advance from Pilot 2 to Pilot 3 with a current Skill Total of 3 would
take (three, plus three) six weeks. A character may only train one
skill in a given week.
So, to restate:
To raise a skill (Total Skill levels)+(Skill level being learned) is training (in weeks) to gain a +1 to skill.
Joe Youngman, 866747, Citizen 1 Term, Age 22, Drive (Wheeled) 2, Flyer (grav) 1, Streetwise (0), Melee 0, Steward 0, Trade (Brewing) 0, Computer 0, and Trade (Glassblower) 0, has a total of 3 levels. Gaining a level 0 in a new skill would be 3 weeks. Raising drive Wheeled to 3 would be 3+3=6 weeks.
Benny Beenthere, Gun Combat (Slug Rifle) 4, Engineering 1, Leadership 1, Athletics (Dodge) 1, Battle Dress 1, Tactics (ground) 2, Heavy Weapons (Launchers) 2, Gun Combat (Energy Pistol) 3, Pilot (Grav) 1, Stealth 2, Computer 1. Total 19 skill levels. It takes him 19 weeks to add a new skill at level 0; he wants to learn to Astrogate. 19 weeks of training later, he's a level 0 astrogator. 20 weeks after that of training, he's added Astrogation 1.
The GM controls the rate of advancement by how he/she defines what constitutes a "week" of training. Some require it to be a week of instruction by a competent teacher. Others allow full time employment to count as some fraction of a week in that skill. Others allow a week to be 40 hours of dedicated skill tapes study and/or practice.
The Rules are intentionally unclear on exactly what constitutes that "week of training."
There are no mechanics for adding contacts in play, no mechanics for adding cash in play - both fall equally into the realm of "GM Fiat" - every bit as much as what constitutes a week of training.