Mathematica has some very powerful string manipulation commands. Try the following:

In[1]:= modifyString[str_String] := 
 ToLowerCase[StringReplace[str, Except[LetterCharacter] -> ""]]

In[2]:= modifyString["I am trying to write a function in Mathematica \
that takes a string and removes any spaces and punctuations and also \
turns capital letters into their lowercase equivalents, and keeps all \
lower case letters. 

 This is what I have so far, and it doesn't seem to work."]

Out[2]= "iamtryingtowriteafunctioninmathematicathattakesastringandremo\
vesanyspacesandpunctuationsandalsoturnscapitallettersintotheirlowercas\
eequivalentsandkeepsalllowercaselettersthisiswhatihavesofaranditdoesnt\
seemtowork"

If you want to allow digits as well then change LetterCharacter above to WordCharacter.

Note that in the above input and output the backslash characters are just a result of the copy and past from Mathematica into this forum. They are not part of the input or the output.