If you want just to correct your code, you can start by using the .lower() method in Python. Using it prevents from your code to get too repetitive (using 'c' or 'C' to get the same string) and from bugs as well.
As for the bugs, your code had three of it. The first one, is that if you still don't want to use .lower(), you must write the or statement like this:
example = 's'
if example == 's' or example == 'S':
print(example)
This is the correct way to use the or statement.
The second bug is at your while loop:
while i <= len(lst):
if j+len(word_to_find) <= len(lst[i]):
if lst[i][j] == 'C' or 'c':
if lst[i][j+1] == 'a'or'A':
if lst[i][j+2] == 't'or'T':
print(i)
j+=1
i+=1
You must remember that the len() function returns the number of elements inside the list or string. As we start counting the position of values at 0 in a object, the last index will be len(lst)-1. For example:
testList = ['a', 'b', 'c']
len(testList) # returns 3
testList[2] # returns 'c'
As you can see, even if I type testList[2], it returns the last value of the list. Your while try to access the lst[5], since it has the less than or equal sign [while <= len(lst)]. Changing it to the less than (<) sign solves the problem.
The last bug I found is your variable j. In your code, it just get added by 1 everytime the loop starts over. The problem is, your next word may not have that specific letter. For example:
lst = ['lion', 'cat']
In your while loop, Python will return a bug for the second word. Cat doesn't have a fourth letter as lion does. To prevent this, you must set the variable j to 0 everytime the loop starts again. You can do this by adding a j = 0 right after your while statement, as below.
lst = ['caT.', 'Cat', '.cat', 'NONE', 'caty']
word_to_find = 'cat'
i = 0
while i < len(lst):
j = 0
if j+len(word_to_find) <= len(lst[i]):
if lst[i][j].lower() == 'c':
if lst[i][j+1].lower() == 'a':
if lst[i][j+2].lower() == 't':
print(i)
j+=1
i+=1
The code above is the final and working form, it will find any string that starts with 'cat'. There is more efficent ways to code a search algorithm like that, but I think you just want help with the bugs in your code. I hope that this help you. Keep coding!