How to write "good" Julia code when dealing with multiple types and arrays (multiple dispatch)

Julia compiles a specific version of your function for each set of inputs as required. Thus to answer part 1, there is no performance difference. The parametric way is the way to go.

As for part 2, it might be a good idea in some cases to write a separate version (sometimes for performance reasons, e.g., to avoid a copy). In your case however you can use the in-built macro @vectorize_1arg to automatically generate the array version, e.g.:

function mysquare{T<:Number}(x::T)
    return(x^2)
end
@vectorize_1arg Number mysquare
println(mysquare([1,2,3]))

As for general style, don't use semicolons, and mysquare(x::Number) = x^2 is a lot shorter.

As for your vectorized mysquare, consider the case where T is a BigFloat. Your output array, however, is Float64. One way to handle this would be to change it to

function mysquare{T<:Number}(x::Array{T,1})
    n = length(x)
    y = Array(T, n)
    for k = 1:n
        @inbounds y[k] = x[k]^2
    end
    return y
 end

where I've added the @inbounds macro to boost speed because we don't need to check the bound violation every time — we know the lengths. This function could still have issues in the event that the type of x[k]^2 isn't T. An even more defensive version would perhaps be

function mysquare{T<:Number}(x::Array{T,1})
    n = length(x)
    y = Array(typeof(one(T)^2), n)
    for k = 1:n
        @inbounds y[k] = x[k]^2
    end
    return y
 end

where one(T) would give 1 if T is an Int, and 1.0 if T is a Float64, and so on. These considerations only matter if you want to make hyper-robust library code. If you really only will be dealing with Float64s or things that can be promoted to Float64s, then it isn't an issue. It seems like hard work, but the power is amazing. You can always just settle for Python-like performance and disregard all type information.

As of Julia 0.6 (c. June 2017), the "dot syntax" provides an easy and idiomatic way to apply a function to a scalar or an array.

You only need to provide the scalar version of the function, written in the normal way.

function mysquare{x::Number)
    return(x^2)
end

Append a . to the function name (or preprend it to the operator) to call it on every element of an array:

x = [1 2 3 4]
x2 = mysquare(2)     # 4 
xs = mysquare.(x)    # [1,4,9,16]
xs = mysquare.(x*x') # [1 4 9 16; 4 16 36 64; 9 36 81 144; 16 64 144 256]
y  = x .+ 1          # [2 3 4 5]

Note that the dot-call will handle broadcasting, as in the last example.

If you have multiple dot-calls in the same expression, they will be fused so that y = sqrt.(sin.(x)) makes a single pass/allocation, instead of creating a temporary expression containing sin(x) and forwarding it to the sqrt() function. (This is different from Matlab/Numpy/Octave/R, which don't make such a guarantee).

The macro @. vectorizes everything on a line, so @. y=sqrt(sin(x)) is the same as y = sqrt.(sin.(x)). This is particularly handy with polynomials, where the repeated dots can be confusing...