Retina, 45 bytes

i`\b((\w)\w*)\b(?<=\b\2\w*\b.+)

\W+
 
^ | $

Simply uses a single regex to remove later words starting with the same \w character (case insensitive with the i option), converts runs of \W to a single space, then removes any leading/trailing space from the result.

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Edit: See @Kobi's answer for a shorter version using M!`

Retina, 28 bytes:

M!i`\b(\w)(?<!\b\1.+)\w*
¶
 

• M! - Match each work and print all words separated by newlines.
• i - Ignore case.
• \b(\w) - Capture first letter of each word
• (?<!\b\1.+) - After matching the letter, check if there wasn't a previous word starting with the same letter. \1.+ ensures at least two characters, so we are skipping the current word.
• \w* - match the rest of the word.
  The above matches only words - all other characters are removed.
• ¶\n - Replace newlines with spaces.

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JavaScript (ES6), 73 71 bytes

s=>s.match(u=/\w+/g).filter(w=>u[n=parseInt(w[0],36)]?0:u[n]=1).join` `

Saved 2 bytes thanks to @edc65!

Test

var solution = s=>s.match(u=/\w+/g).filter(w=>u[n=parseInt(w[0],36)]?0:u[n]=1).join` `;
var testCases = [
  "Ferulas flourish in gorgeous gardens.",
  "Take all first words for each letter... this is a test",
  "Look ^_^ .... There are 3 little dogs :)",
  "...maybe some day 1 plus 2 plus 20 could result in 3"
];
document.write("<pre>"+testCases.map(t=>t+"\n"+solution(t)).join("\n\n")+"</pre>");

Pyth, 21 bytes

J:z"\w+"1jdxDJhM.ghkJ

Try it online: Demonstration or Test Suite

J:z"\w+"1 finds all the words in the input using the regex \w+ and stores them in J.

.ghkJ groups the words by their first letter, hM takes the first from each group, xDJ sorts these words by their index in the input string, and jd puts spaces between them.

05AB1E, 40 bytes

Code:

94L32+çJžj-DU-ð¡""Kvy¬Xsl©åï>iX®«Uy}\}ðý

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Julia, 165 155 151 129 102 bytes

g(s,d=[])=join(filter(i->i!=0,[(c=lcfirst(w)[1])∈d?0:(d=[d;c];w)for w=split(s,r"\W",keep=1<0)])," ")

This is a function that accepts a string and returns a string.

Ungolfed:

function g(s, d=[])
    # Split the string into an array on unwanted characters, then for
    # each word, if the first letter has been encountered, populate
    # this element of the array with 0, otherwise note the first letter
    # and use the word. This results in an array of words and zeros.
    x = [(c = lcfirst(w)[1]) ∈ d ? 0 : (d = [d; c]; w) for w = split(s, r"\W", keep=1<0)]

    # Remove the zeros, keeping only the words. Note that this works
    # even if the word is the string "0" since 0 != "0".
    z = filter(i -> i != 0, x)

    # Join into a string and return
    return join(z, " ")
end

Saved 53 bytes with help from Sp3000!

Jelly, 32 31 bytes

ØB;”_
e€¢¬œṗf€¢¹ÐfµZḢŒlQi@€$ịj⁶

Try it online!

Vim 57 keystrokes

:s/[^a-zA-Z_ ]//g<cr>A <cr>ylwv$:s/\%V\c<c-v><c-r>"\h* //eg<c-v><cr>@q<esc>0"qDk@q

Explanation:

:s/[^a-zA-Z_ ]//g                                 #Remove all invalid chars.
A <cr>                                            #Enter insert mode, and enter 
                                                  #a space and a newline at the end
ylwv$:s/\\c%V<c-v><c-r>"\h* //eg<c-v><cr>@q<esc>  #Enter all of this text on the 
                                                  #next line

0                                                 #Go to the beginning of the line
"qD                                               #Delete this line into register
                                                  #"q"
k@q                                               #Run "q" as a macro  

#Macro
ylw                                               #Yank a single letter
   v$                                             #Visual selection to end of line
     :s/                                          #Substitute regex
       \%V\c                                      #Only apply to the selection and 
                                                  #ignore case
            <c-v><c-r>"                           #Enter the yanked letter
                       \h*                        #All "Head of word" chars
                                                  #And a space
                           //                     #Replace with an empty string
                             eg                   #Continue the macro if not found
                                                  #Apply to all matches
                               <c-v><cr>          #Enter a <CR> literal
                                        @q<esc>   #Recursively call the macro

I'm really dissapointed by how long this one is. The "Invalid" chars (everything but a-z, A-Z, _ and space) really threw me off. I'm sure there's a better way to do this:

:s/[^a-zA-Z_ ]//g

Since \h matches all of that expect for the space, but I can't figure out how to put the metachar in a range. If anyone has tips, I'd love to hear em.

Lua, 172 Bytes

It ended up way longer that I wanted...

t={}(...):gsub("[%w_]+",function(w)b=nil for i=1,#t
do b=t[i]:sub(1,1):lower()==w:sub(1,1):lower()and 1 or b
end t[#t+1]=not b and w or nil end)print(table.concat(t," "))

Ungolfed

t={}                           -- initialise the accepted words list
(...):gsub("[%w_]+",function(w)-- iterate over each group of alphanumericals and underscores
  b=nil                        -- initialise b (boolean->do we have this letter or not)
  for i=1,#t                   -- iterate over t
  do
    b=t[i]:sub(1,1):lower()    -- compare the first char of t's i word
       ==w:sub(1,1):lower()    -- and the first char of the current word
           and 1               -- if they are equals, set b to 1
           or b                -- else, don't change it
  end
  t[#t+1]=not b and w or nil   -- insert w into t if b isn't set
end)

print(table.concat(t," "))     -- print the content of t separated by spaces

C# (LINQPAD) - 136 bytes

var w=Util.ReadLine().Split(' ');var l=string.Join(" ",w.Select(s=>w.First(f=>Regex.IsMatch(""+f[0],"(?i)"+s[0]))).Distinct());l.Dump();