That's going to be something I need either an insane amount of time and luck on my hands, or eleventy billion mystical essences for, right?

Or is there a secret to getting, for example, a 5-Vis?

To breed for mutations, you just need parents who either show or carry mutations (/genes)
You can catch them, buy them, generate them, or breed with other people's pets to achieve this.

If a pet is listed as Tamed, a single Genetic mutator can be used on them (Lesser mutators add one random trait, either visible or carried, into a completely unoccupied genetic slot (that is, 'NN'), whilst a Greater mutator allows a CHOSEN trait to be added to an unoccupied genetic slot)

If you're looking for specific pets with specific traits, carried or visible, the Advanced Search's "Search Pets" function lets you choose species, if it has traits (Any/Visible/Carried/Either/Nothing options for all gene and mut slots, and the ability to select either Any or a specific trait), and can sort those for sale in ascending price for breeding or sale.

Mystical essences are basically highly-colour-accurate versions of Enhanced Essences, and give the same 4GP amount, so if colouration is not an issue, you'd want to go for the cheaper Enhanced Essences.

You could take a look at "Xavi's 6-vis Breeding Guide!", which goes over this kind of thing in far more depth, but as a general 'vis and carried' starting point...

Let's use our Leihu and Gourami as an example pairing
x

As you can see, the only visible mutation between them is Leihu's Unicorn Tail (Tips)

Their genetics are as such ('N' being the site's 'code letter' for 'nothing', and is dominant over genes and muts):
Leihu:
gd1=BD
gd2=AB
gd3=AC
md1=FH
md2=NN
md3=NA
Gourami:
gd1=DF
gd2=BC
gd3=AB
md1=NG
md2=NE
md3=NA

Which means that ANY offspring they produced would have 3 visible Genes, with a 50/50 chance that they'd have a visible tail mutation (combination possibilities NF, NG, FG, GH- FG being vis Unicorn Tail (Tips) and GH being Snow Leopard Tail), a 1/2 chance at having carried Jingle Bells mutation (md2=NE) and a 1/4 chance of a visible Feathered Wings mutation with 50/50 chance of it being carried instead.

gd1= BD x DF
BB, BF, DD, DF
(50% either Tiger or Marbled vis)
gd2= AB x BC
AB, AC, BC, BB
(50% either Wing Tips or Faded Tips vis)
gd3= AC x AB
AA, AB, AC, BC
(75% Frost Runes, 25% Tribal Runes vis)
md1= FH x NG
NF, NH, GH, FG
(50% no-vis (25% carried Unicorn Tail (Tips) 25% carried Snow Leopard Tail (Tiger)), (25% visible Unicorn Tail (Tips), 25% visible Snow Leopard Tail
md2= NN x NE
NN, NN, NE, NE
(50% no-vis, no-carry, 50% carried Jingle Bells)
md3= NA x NA
NN, NA, NA, AA
(75% no-vis (25% no-vis, no-carry, 50% carried Feathered Wings), 25% Feathered Wings vis)

(which means there is, in fact, a small chance for a 5-vis offspring with 1 carry, actually, despite the parents only having a single visible mutation between them, eg.
)

The Sylesti genetics are simple punnet square type, with N being most dominant, and the rest of the alphabet following A-through-Z in descending order.

Don't know if we helped any, but we think it should help a bit, at any rate.

Lol yay, Scathy coming in wth the clutch man XD

Okay then. There's no way I'm going to be able to breed for a 3-Vis with my current pets, since all my pets have their Vis mutations in different slots.

You can breed a 3-vis with a series of pets that have mutations in different slots.

If you have A with M1 vis, B with M2 vis and C with M3 vis then you'd do something like this:

A breeds with B and produces offspring AB with M1 carry and M2 carry
You'd then breed AB with C and ideally get an offspring ABC with M1 carry, M2 carry and M3 carry
You'd then breed ABC with either A or B and ideally get an offspring with M1 carry or visible, M2 carry or visible and M3 carry
Continue until you have your 3 vis

The ideally's above are the best possible outcomes from breeding the two pets together so there will be trial and error as breeding pets with carries can, at worse, lead to offspring with no trait points and, at best, lead to offspring that have all carries passed on from both parents.

Edit: In order to run projects like this, you're best to get each offspring tested so you know where you stand in relation to its carries.