At 1st Level:
You start with at least 4 skill proficiencies; at least 2 from your class + 2 from background.
Rogues start with 6 total. Bards and Rangers start with 5 in total.
Races
Half Elves get an extra two skills and variant rules human start with an extra skill.
Elves start proficient in perception.
Half Orcs start proficient in intimidation.
After Level 1:
Yep, no method of training for skills in the default rules.
Training in the PHB only applies to Tools proficiencies or languages.
Optional Rules can grant skills:
Feats Skilled feat gives proficiency in three extra skills.
Multiclassing into Ranger, Rogue or Bard gives you an extra skill from their class skill lists (so any for Bard).
As for further specialisation Rogues and Bards get Expertise that lets then add their proficiency bonus again to a few of their skills.