Most keys do the same thing in Picture mode that they usually do, but do it in a quarter-plane style. For example, C-f is rebound to run picture-forward-column, a command which moves point one column to the right, inserting a space if necessary so that the actual end of the line makes no difference. C-b is rebound to run picture-backward-column, which always moves point left one column, converting a tab to multiple spaces if necessary. C-n and C-p are rebound to run picture-move-down and picture-move-up, which can either insert spaces or convert tabs as necessary to make sure that point stays in exactly the same column. C-e runs picture-end-of-line, which moves to after the last nonblank character on the line. There is no need to change C-a, as the choice of screen model does not affect beginnings of lines.
Insertion of text is adapted to the quarter-plane screen model through the use of Overwrite mode (Section 32.1). Self-inserting characters replace existing text, column by column, rather than pushing existing text to the right. RET runs picture-newline, which just moves to the beginning of the following line so that new text will replace that line.
Picture mode provides erasure instead of deletion and killing of text. DEL (picture-backward-clear-column) replaces the preceding character with a space rather than removing it; this moves point backwards. C-d (picture-clear-column) replaces the next character or characters with spaces, but does not move point. (If you want to clear characters to spaces and move forward over them, use SPC.) C-k (picture-clear-line) really kills the contents of lines, but does not delete the newlines from the buffer.
To do actual insertion, you must use special commands. C-o (picture-open-line) creates a blank line after the current line; it never splits a line. C-M-o (split-line) makes sense in Picture mode, so it is not changed. C-j (picture-duplicate-line) inserts below the current line another line with the same contents.
To do actual deletion in Picture mode, use C-w, C-c C-d (which is defined as delete-char, as C-d is in other modes), or one of the picture rectangle commands (Section 27.4).