How would you get from Penn Station to East 23rd Street and first Avenue by subway?

There is no convenient way to get there by subway, because you need to go pretty much straight across town, and there are very few subway lines that run crosstown.

I would just walk or take a cab. It's only a little over 1.5 miles (2.4 km): 10 or 11 blocks south to 23rd Street, then 8 or 9 avenue blocks east to 1st Avenue. Penn Station is at 33rd to 34th Street, between 8th and 7th Avenues. Actually, technically it's 9 or 10 blocks east, counting Broadway, but Broadway runs diagonally north/south and cuts between avenues, so the blocks it runs between are not full length.

If you really want to take the subway, then you can take the Uptown E train (of the A-C-E, the blue line) from 34th Street-Penn Station to Lexington Avenue-53rd Street. This station is connected to the 51st Street Station on the 6train (of the 4-5-6, the dark green line). Take the Downtown 6 to 23rd Street (at Park Avenue South, 4 blocks east of 1st Avenue).

There's another route you can take. You'd have to make 2 transfers, but there'd be fewer stops, and you wouldn't be taken as far out of your way. You can take the Uptown 1-2-3 (the red line), 1 stop, from 34th Street-Penn Station to 42nd Street-Times Square.

Transfer at Times Square to the S shuttle train (color grey) east across 42nd Street, from Times Square (West 42nd) to Grand Central Station (East 42nd). At 42nd Street-Grand Central Station, transfer to the Downtown 6 to 23rd Street.