I learned to read music to a certain level as a child and since retiring have started learning piano again. But no matter how many books I read, I can't figure out, when reading sheet music, when you are supposed to cross hands, or what I am supposed to play and with what hand when there is a treble sign on the bass clef staff, or bass clef sign on the treble staff. I know anything on the bass clef stave should be played with the left hand, and the treble with the right, but am confused when things don't fall neatly into their respective places. Help!
I would say three things:
System position. Basically people think bass clef for the left hand, treble clef for the right hand. But, clefs can change. The grouping of two staves makes a system for the grand staff. Rather than clef type, it's the system position that matters. The lower staff is for the left hand and the upper staff is for the right hand.
...in that example the left hand is the lower staff and the clef changes are used to keep the notes on the staff with fewer ledger lines.
Hand abbreviations. L.H. for left hand and R.H. for right hand (or m.s. and m.d.) will label parts of the notation when the notation move the hands between the usual staves...
...that was from the Mozart Nannerl Notebook.
Stem direction. In some cases stems up mean right hand and stems down mean left hand...
Keep in mind stem direction is also used to show multiple voices played by one hand...
...so, stem direction doesn't always mean left/right hands.
Basically all notes shown are played with r.h. if written on treble clef, and notes shown on bass clef are played with l.h. When there are notes written using ledger lines, they're there to indicate which hand plays - treble - r.h. , bass - l.h.
That's about it really. Otherwise there would be little point in having the two separate staves. Even when there are two treble (or bass) clefs, the right hand plays the top staff, left hand , the bottom.
The right hand plays the upper staff, and the left hand plays the lower staff, regardless which clefs are displayed. The clefs only tell you how to interpret the lines and spaces; they don't tell you which hand to use.
When playing duets, for example, it's very common that one part will have both hands indicated in treble clef and the other both hands in the bass clef.
Edit: As Andrew Chin mentioned, sometimes there will be an explicit instruction as to which hand to use. Otherwise, the above holds.
An example of explicit instructions for which hand plays which notes, see What does the L-shaped symbol attached to C5 and G4 on the top staff mean?