removing duplicates from a ruby array without using built in methods... eg. .uniq, .sort etc

I don't know which built-in methods you can't use, but if you can use Array#| it's hard to imagine anything simpler than

array | []
  #=> [1, "fred", 6, 5, "john", "peter"]

Note this retains the ordering of array.

array & array works as well.

Creating and returning a secondary cleaned array makes this fairly simple: iterate over the original array and append only those elements that do not already exist within the cleaned array:

def clean(array)
  cleaned = []
  array.each do |element|
    cleaned << element  if not cleaned.include?(element)
  end
  return cleaned
end

This is relatively quick and does exactly what you want.

Proof of concept (irb):

2.2.2 :001 > def clean(array); cleaned=[];  array.each{|ele| cleaned << ele  if not cleaned.include? ele}; cleaned; end
 => :clean

2.2.2 :002 > clean [1, "fred", 6, 5, "fred", "john", 6, "peter", "fred"]
 => [1, "fred", 6, 5, "john", "peter"]

You could improve processing times on large arrays by using a Hash instead. This would work by appending elements as keys to improve the internal lookup times, and then simply return a list of its keys for your array. (Caveat: doing this may return them in an arbitrary order, however, so further effort may be required to maintain the original order.)

Edit: I missed the sorting requirement in your expected output. Apologies.

Building on what @terra did:

def uniq(array)
  uniq = []
  array.each do |element|
    uniq << element unless uniq.include?(element)
  end
  uniq
end

def sort(array)
  sorted = []
  [Integer, String].each do |type|
    sorted += array.select { |element| element.is_a? type }
  end
  sorted
end

So now if you do this:

p uniq_and_sorted = sort(uniq([1, "fred", 6, 5, "fred", "john", 6, "peter", "fred"]))

You should get:

[1, 6, 5, "fred", "john", "peter"]

This maybe long winded, but it works:

    array = [1, "fred", 6, 5, "fred", "john", 6, "peter", "fred"]

    numbers_array = []
    array.each do |e|
      numbers_array << e if e == e.to_i
    end

    numbers_array.sort!

    strings_array = []
    array.each do |e|
        strings_array << e if e == e.to_s
    end

    strings_array.sort!

    final_array = []
    final_array << numbers_array << strings_array
    final_array.flatten!
    final_array.uniq!

    p final_array