I've just started Python and need to write a program that tells if a string is a palindrome. Here is what I have so far.

def palp(word):
    if len(word) < 2:
        return True
    left_index =  (0)
    right_index = len(word) - 1
    while len(left_index) <> len(right_index):
        return False
    if left_index == right_index:
        left_index + 1
        right_index - 1
        return True

I don't know but for some reason it doesn't return true for any palindrome except for ones with only one letter.

You can not take len of integer values, wouldn't you need to check letters in word? You also forgot CODE tags.

def palindromic(s):
    return True if str(s) == str(s)[::-1] else False

Unnecessary complicated, but forgetting case of word differences.

This simple version, which does not deal with palindromic sentences:

def palindromic(s):
    s = s.lower()
    return s == s[::-1]
>>> palindromic('Saippuakivikauppias')
True

As pyTony say you complicated it by use of ternary operator( a if b else c )
One thing also to remember is to remove punctuation.
Here one with translate.

from string import punctuation

def is_palindrom(word):
    word = word.lower().translate(None, punctuation)
    return word == word[::-1]

print is_palindrom('Saippuakivikauppias') #True

And a test with a longer palindromic sentence.

def is_palindrome(s):
    letters = [c for c in s.lower() if c.isalpha()]
    return letters == letters[::-1]

s = '''\
A man, a plan, a canoe, pasta, heros, rajahs, a coloratura, maps, snipe,
percale, macaroni, a gag, a banana bag, a tan, a tag, a banana bag again
(or a camel), a crepe, pins, Spam, a rut, a Rolo, cash, a jar, sore hats,
a peon, a canal ? Panama!'''

print is_palindrome(s) #True
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