What does a nucleus get out of the process of decay? Why does it bother?
The answer is that the nucleus becomes more stable. Systems become more stable when they go from a high energy arrangement to a lower energy one. This means the system releases energy.
For example a domino on its end is unstable. Give it a little push and it ends up flat on the table. The centre of gravity has got lower and so the energy of the system has decreased. This energy appears as vibrations of the air (i.e. sound), of the table and of the domino. The vibrations rapidly become smaller-scale and less ordered, ending up as internal energy of the atoms. So the temperature rises a little.
You can model a nucleus as two upright wooden blocks standing on top of each other. This little tower is quite unstable. Give it a little push and it falls over. There are three ways of making the tower more stable that are analogous to alpha, beta and gamma decay.
For alpha decay, simply remove the top block. Making a big nucleus a little smaller increases its stability.
For beta decay, replace the top upright block with a shorter one. Changing a nucleon from a neutron into a proton can make a nucleus more stable.
For gamma decay, place the top block on its side so the blocks stay the same but the configuration is different. Letting nucleons settle down into a lower energy pattern makes the nucleus more stable.
Here is an interactive animation modelling alpha, beta and gamma decay with wooden blocks.