When you 'lay the foundations' for something it means to create a base upon which other structures or larger ideas are built. You lay foundations before building a house for example or you could say that Darwin's theory of evolution laid the foundations for modern evolutionary theory. To 'lay the groundwork' has a similar meaning of creating a foundation for something or to put in the necessary work for something to begin. 

1. The restructuring of the company last year is what has laid the foundation for the success we are currently enjoying.
2. Rebecca credited her year-long apprenticeship at the company with laying the foundation of her future career in the industry.
3. Last week's two hour meeting between the rival executives helped to lay the groundwork for the deal that was signed this morning.

1. I have laid the groundwork for the garden and hope to start planting in the Spring.
2. My mother laid the foundations for my interest in music by taking me to concerts when I was a child.