[RPG] What skill proficiencies can a Variant Human choose from at character creation

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.

No, you cannot stack proficiencies.

You're either proficient with a skill/item/weapon and get to apply your proficiency bonus, or you're not; you can't be doubly or triply proficient in something. Some class features and special abilities may allow you to double your proficiency bonus in some situations, but these are always explicitly spelled out. As the PHB states on page 12, under the "Proficiency Bonus" heading:

Your proficiency bonus can't be added to a single die roll or other number more than once. Occasionally, your proficiency bonus might be modified (doubled or halved, for example) before you apply it. If a circumstance suggests that your proficiency bonus applies more than once to the same roll or that it should be multiplied more than once, you nevertheless add it only once, multiply it only once, and halve it only once.

And this is reiterated in further detail in the "Proficiency Bonus" section on page 173:

Your proficiency bonus can't be added to a single die roll or other number more than once. For example, if two different rules say you can add your proficiency bonus to a Wisdom saving throw, you nevertheless add the bonus only once when you make the save.

Occasionally, your proficiency bonus might be multiplied or divided (doubled or halved, for example) before you apply it. For example, the rogue's Expertise feature doubles the proficiency bonus for certain ability checks. If a circumstance suggests that your proficiency bonus applies more than once to the same roll, you still add it only once and multiply or divide it only once.

If you would be granted the same tool/skill proficiency by your background and your class during character generation, you get to just choose a different proficiency of the same type.

Extra: However, as mattdm's answer points out, Xanathar's Guide to Everything introduces a new feat, Prodigy, which amongst other things allows you to choose one skill you have proficiency in and double your proficiency bonus when using that skill - so as a variant human taking a feat at 1st level, you could improve your bonus to one skill in this way.