For me, I use the terms slow food and local food almost interchangeably. I don't know if the following explanation is technically what slow food is supposed to me but this is what I take it as.
We started making a real effort to eat locally/slowly. The benefits I see are: less transport or the food saving on fuel costs/emissions, fresher and healthier food b/c it was not 'bred for transport', supporting your local small farmers and economy, and supporting sustainable ag-practices. It's getting to know where and who your food comes from.
We recently joined out local CSA. I've found that prices on about half of the items are comparable to the organics in the grocery store, while some of the items are a bit more expensive. In the summer, we grow our own veggie garden and supplement with produce from the farmers market. We've also been trying to find direct sources of different local foods to offset the co-op fees.
I also thin a big part of slow-food is eating what is in season. Trying to eat what nature has to offer when it has to offer or preserving it (canning, freezing) to be enjoyed later.
We are making small steps and are no-where near where we ideally like to be with our eating. We did make a conscious effort to forgo fresh tomatoes in winter, unless they are coming from a local greenhouse. On the other hand, coffee is not grown anywhere around here and for now it is something we deem a necessity so we buy organic/fair trade. baby steps
