how to find the index of a repeated character in lua string

There are several ways to do this. Perhaps the simplest is using the () pattern element which yields a match position combined with string.gmatch:

for index in ("/home/user/dev/project"):gmatch"()/" do
    print(index)
end

which prints

1
6
11
15

as expected. Another way to go (which requires some more code) would be repeatedly invoking string.find, always passing a start index.

Assuming that you probably want to split a string by slashes, that's about as simple using string.gmatch:

for substr in ("/home/user/dev/project"):gmatch"[^/]+" do
    print(substr)
end

(the pattern finds all substrings of nonzero, maximal length that don't contain a slash)

Documentation for patterns is here. You might want to have a look at the subsection "Captures".

There are many ways to do so.
Also its good to know that Lua has attached all string functions on datatype string as methods.
Thats what @LMD demonstrates with the : directly on a string.
My favorite place for experimenting with such complicated/difficult things like pattern and their captures is the Lua Standalone Console maked with: make linux-readline
So lets play with the pattern '[%/\\][%u%l%s]+'

> _VERSION
Lua 5.4
> -- Lets set up a path
> path='/home/dev/project/folder with spaces mixed with one OR MORE Capitals in should not be ignored'
> -- I am curious /home exists so trying to have a look into
> os.execute('/bin/ls -Ah ' .. ('"%s"'):format(path:match('[%/\\][%u%l%s]+')));
knoppix  koyaanisqatsi
> -- OK now lets see if i can capture the last folder with the $
> io.stdout:write(('"%s"\n'):format(path:match('[%/\\][%u%l%s]+$'))):flush();
"/folder with spaces mixed with one OR MORE Capitals in should not be ignored"
> -- Works too so now i want to know whats the depth is
> do local str, count = path:gsub('[%/\\][%u%l%s%_%-]+','"%1"\n') print(str) return count end
"/home"
"/dev"
"/project"
"/folder with spaces mixed with one OR MORE Capitals in should not be ignored"

4
> -- OK seems usefull lets check a windows path with it
> path='C:\\tmp\\Some Folder'
> do local str, count = path:gsub('[%/\\][%u%l%s]+','<%1>') print(str) return count end
C:<\tmp><\Some Folder>
2
> -- And that is what i mean with "many"
> -- But aware that only lower upper and space chars are handled
> -- So _ - and other chars has to be included by the pattern
> -- Like: '[%/\\][%u%l%s%_%-]+'
> path='C:\\tmp\\Some_Folder'
> do local str, count = path:gsub('[%/\\][%u%l%s%_%-]+','<%1>') print(str) return count end
C:<\tmp><\Some_Folder>
2
> path='C:\\tmp\\Some-Folder'
> do local str, count = path:gsub('[%/\\][%u%l%s%_%-]+','<%1>') print(str) return count end
C:<\tmp><\Some-Folder>
2