I see many similar posts, but none specific enough to my situation for this rookie to be able to apply the answers.

I have box with a light switch on one side and a standard outlet (i.e. I can plug two things into it) on the other. I need to replace the outlet with a GFCI.

There are four wires coming into the box, from two different places. The top left of the box has a white and a black. The top right of the box has a white and a red. I do not know which side is in and which side is out. I do not think that I need to make any considerations for what is further down the line, i.e for what else is actually protected by the GFCI.

Either via bravery or stupidity, I killed the fuse from the breaker and have made a few attempts. The existing outlet had four lines coming into it. I swapped in the GFCI, putting the two whites on the side indicated on the outlet itself, and the other two on the other side.

Switched the fuse back on. Saw small green indicator light on the GFCI. It would not test or reset, and I couldn't turn on the light. Instructions with the GFCI indicated that I had screwed up line vs. load. I tried again, swapping white for white, and black for red. Same result. :-(

Any advice?

Can we assume that you did not disturb the switch?

The red is suspiciuos because there is usually a black white and red in the cable.

GFCI's have an in and an out. It also depends if you WANT to protect outlets downstream.

Green means the outlet is working. Test should make the green light go out.

On the old outlet, was the connection between the brass terminals removed?

Suggest this for starters: Both whites together. The red and black together. White to Line side of GFCI (white terminal). The red/black to the load side (brass colored terminal

If you read the instructions you'd see that there are LINE and LOAD terminals. You should have used only the LINE terminals.

What you did was wire it using both line and load and you put the feed on the load side which will show power but the device will not work. Newer GFIs will not work if miswired.

Put the two whites on the silver LINE screw and the two blacks on the gold LINE screw.

I have seen GFI's Show the LED when working, other GFI's show the LED when Tripped. And the latest ones I have seen Don't light when Good or Tripped, It is now an "End of Life Indicator".
I don't like them.