I have been reading the Basic Rules a couple times, but could not find any way to gain proficiency in skills past level 1.
For example, if I create a Rogue, I may chose 4 skills among 11:
Acrobatics, Athletics, Deception, Insight, Intimidation, Investigation, Perception, Performance, Persuasion, Sleight of Hand and Stealth.
Now, a Criminal background grants proficiency in Deception and Stealth already, so I can pick 4 among 9:
Acrobatics, Athletics, Insight, Intimidation, Investigation, Perception, Performance, Persuasion and Sleight of Hand
which still leaves 5 of them on the floor.
I could find two nuggets:
The Human Variant Trait seems to allow to get proficiency in one skill.
The multi-classing rules (Chapter 6) seem to imply that one may get some proficiencies from multi-classing; though they refer to the PHB and do not specifically imply skill proficiencies (could be weapon or armor or tools…)
Are there other sources of skills that would allow one to broaden one's range of skills?
There are a few classes you can multiclass into to gain skills, such as Bard, Ranger or Rogue.
In addition to that, in the player's handbook there is a feat called "Skilled" which allows you to gain proficiency in any combination of 3 skills or tools.
In Xanathar's Guide to Everything, we have the Prodigy feat which gives a skill proficiency and a few other things, but can only be taken by a Human, Half-Orc, or Half-Elf.
In Tasha's Cauldron of Everything, the feat Skill Expert gives proficiency in a skill and a few other benefits, with no racial prerequisite.
There is also a bard ability in the college of lore that allows you to learn 3 more skills at level 3.
Currently, the downtime rules allow you to gain proficiency in tools but not skills.
In case it isn't clear, you don't have to pick skills from your class skill list when you gain new skills through the "Skilled" feat. You can pick any skill you want.