Try it online!
The first 13 plus signs put the 'newline' character into cell 0. Then, I initialize cell 1 as a counter with a value of 3. Then there's the main loop. This loop will get a character in cell 2, display it, display the 'newline' character in cell 0, add cell 2 to cell 3, which holds the running sum, and repeat until cell 1 has a value of zero. When the loop is done, it displays the value in cell 3, which, if you've been following, holds the running sum of the characters. However, there is a problem with this last part: it displays the sum of the three characters as an ascii character, not an integer. I don't know if this is actually an issue since I technically displayed the sum, but just in case, I added a (long) routine to the end of the code that displays the sum as an integer.
Try it online!
But wait, there's more. The routine I used only supports displaying eight bit numbers. This means that if I input 'abc' into the program, it displays:
If you didn't catch it, the sum of the ascii values of 'a','b', and 'c' is not 39. However, it's equivalent if you add 255. This is because the value of the cell was only calculated to 8 bits by the routine I used. At this point, brainfuck had had its effect on my brain, and I didn't really feel like rewriting any more code in it. If I need to change my submission because of this issue or any other issue, let me know in the comments.