I apologise if this has already been posted. I need an answer fairly quickly; I don't have time searching for what I'm after.

Anyway, I have a picture that I'd like to remove the white background from. There is some white in the picture which I don't want to get rid of, but I've tried the ways I already know of (magic wand, load channel as selection, etc.) and they either remove this white too, or they spread into the image (magic wand).

I'd like to know if there are any other ways I can remove this white background without affecting the picture's outline, or cutting some of it off at all, or the white that's supposed to be there.

you can remove the background pixel by pixel with the eraser tool

whats the purpose: what are you planning next?

The pixel-by-pixel step is tedious, and the outline of my picture isn't crisp.

I'd just like it so it has a transparent background, because I'm considering layering it onto another image. And I can't just have a block of white behind the picture, hiding the image behind it.

k,
first try out other layer modes like multiply, darker color etc...
If that doesnt work for you,
try the following:

Paste your image on a seperate layer:
Click to view attachment

Then go to the effects icon in the layer palette:
Click to view attachment

Then play with the blending options... click the white triangle to remove white, alt click the triangle to seperate the 2 halfs so you get softer transitions:
Click to view attachment

Quick and dirty result just to show the effect (didnt tweaked a lot since it was just for demo purposes):
Click to view attachment

Let me know if this worked.

I haven't tried it yet, but I just want to be sure about something. Will that way remove all the white from the image? Because it's just the white background I want to remove. There's white inside the picture which needs to stay there, otherwise I'd have used the ways I know to remove all white.

removes all the white (non-destructive),
but the parts you dont want in the image itself, just mask them out; i reckon the masks dont have to be precise, just protect the white you still need.