In D&D 5e, is the only way to get higher skill ranks determined by prof bonus and ability score? Is my only choice to wait til 4th level, raise an ability score, and improve my proficiency bonus from +2 to +3? Or are there other ways?
Other than ability increase and level there are a few ways to increase your skill modifiers: they are based mostly on class.
For example Bards and Rogues get the Expertise trait which lets them double proficiency bonus on selected skills of the players choice. A Knowledge Domain Cleric can gain double proficiency bonus in certain skills. The Ranger's Natural Explorer feature, Sorceror's Dragon Ancestor, and Dwarf's Stonecunning also double bonuses in narrowly-tailored circumstances.
But beyond a specific grant, there is no way to increase a skill check besides increasing a stat to earn a higher bonus or by leveling up and gaining a higher proficiency modifier.
Everyone is capable of using every skill whether they have proficiency or not. Proficiency just grants them a bonus appropriate to their level.