I play trumpet, and have transposed quite a lot, playing Bb parts on a C trumpet, and the contrary. Also older symphonic parts would notate everything in C and just write "in Eb" when the tonality changed to Eb.
I have only used the clef trick described by @MattPutnam, and wanted to add to it for accidentals. He is right that this strategy applies mostly for monophonic instruments, but I guess it's not impossible to use on keyboards.
Know which accidentals are associated with this specific transposition. Keep any other accidental as they are in the original music. For these "transposition accidentals", you need to either raise or lower half a step compared to the original accidental (even natural).
You want to transpose a tune a fourth up. If the original was in C, you want to play it in F. Find the clef that makes a notated C turn into an F (it's the second C-clef, mezzo-soprano). This means you need to be comfortable reading in other clefs.
The key signature difference between C and F is one added flat. This means that every time you'd play a B according to your new clef, play a Bb. Every time you'd play a Bb, play a Bbb. Everytime you'd play a B#, play a B. You lower the accidental (even natural) of any B.
Another way to look at it is that you change the key signature according to the transposition. Transposing a fourth upwards means adding a flat to the key signature. Assume original tonality of G major, one sharp. Adding a flat to a key signature is the same as removing a sharp, so your "one sharp" key signature of G major becomes "all natural" key signature. Again, the accidentals are unchanged except for all notated F.
- F# -> B natural
- F natural -> Bb
- Fb -> Bbb
(you lower the accidentals whenever you play any kind of B)