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.
You don't get to choose a different save proficiency
The rule you're thinking of that lets you choose a different skill or tool proficiency if you gain a proficiency you already have isn't a general rule, it's a rule only for proficiencies gained from Backgrounds. The rules is on page 125 of the PHB, and is part of the rules on how to apply the benefits offered by Backgrounds. In addition, Backgrounds are only available at level 1 during character creation, so that rule will never apply during later character advancement choices.
Since Slippery Mind and Resilient aren't Backgrounds, if you gain a redundant proficiency from one of them, that rule doesn't apply. And since there's no similar general rule that lets you choose a different proficiency when gained from a non-Background source, getting to choose an alternative proficiency isn't something the game lets you do.
Unfortunately, you'd just gain proficiency in Wisdom again, which would grant the character nothing new.
Best Answer
Yes
Musical instruments are a kind of tool. The Skilled feat lets you gain proficiency with tools. Therefore Skilled lets you gain proficiency with a type of musical instrument.
Note: When picking an instrument, make sure to talk to your DM to ensure you'll actually get to use it. There's no point in proficiency with Pipe Organs if your character never finds a pipe organ to play.