You can argue about this some because there were many different people at the Constitutional Convention and they had different ideas about what the government should be like. People like Alexander Hamilton had a very expansive vision for the federal government while others thought it should be much less powerful.
In general, though, the role the Framers foresaw was more limited than what we now have. The federal government was going to mainly provide two things: defense for the country and a good business climate.
The first of these is obvious, the second less so. What I'm saying is that the Framers wanted to have a strong economy and they thought the national government should make that happen. They thought that it should prevent trade wars between states and that it should maintain a single currency. Many of them hoped it would create infrastructure that would help the country grow.
So, in general, the government was supposed to be a supportive, rather than an instrusive, entity. It was supposed to keep Americans safe and help them have a chance at prosperity.