What are the advantages of a 2560 over a uno, or are there any? Will the shields for the uno work on the 2560?
Main advantage, more program space and memory. But in most cases that isn't really needed.
Some will say the number of IO is an advantage but I disagree. It indeed has more IO but that just makes it easy to turn you're project into a rats nets of wire. So I think it's better to split the project into pieces or make a custom PCB instead of using a Mega.
Most shields will work on a Mega but not all. Some really use a dedicated part of the Uno (being it hardware or software). You might be able to fix it in software. Basically it's like all shields, they all have weird side effects 
The fact that the Mega has 3 spare HardwareSerial ports can be a big advantage for some projects.
...R
A shield will almost certainly work on an UNO unless it specifically says that it is designed for a different Arduino. Many will work on both an UNO and a MEGA. Some will NOT work on a MEGA because they rely on hardware interfaces (I2C, SPI) that are on different pins on the MEGA.
General recommendation: Use the UNO unless you specifically need the capabilities of the MEGA.