Aquiring Sleight of Hand skill for my character

[Mastikator]

As has been said the PHB has training tools and languages as a downtime activity, taking 250 days and gold. The DM can also give additional options.

Xanathar's revise these rules (but Xanathar's is optional), to 10 work weeks, still 250 gold. So only 50 days basically (or 70, depending on how the DM defines a work week in their campaign). RAW it's up to the DM.

An actual skill is worth more than a language or tool. So IMO it should cost more. OR if you are retraining a skill, IE changing from one skill to another, it should cost less. Maybe only 7-8 work weeks?

Another option is to ask the DM if you could craft or buy a customized magic item, the only thing the magic item does is grant proficiency in sleight of hand. Maybe some kind of lesser variant thief's gloves (basically a magic item with a specific minor boon from the artifact list, and nothing else). There are rules for crafting/buying magic items in Xanathar's.
This option is more expensive, and perhaps more dangerous/fun, the DM has IMO more chances to turn this into a mini-adventure.