I don't recommend running a drain directly through a lawn if at all possible. It is best to slope the lawn to create some run-off, place any drainage at the highest point to divert excess moisture and concentrate on improving the friability of the soil conditions.
For drainage to be effective it must be constructed properly. The critical steps are to use the right aggregate and make sure the pipe being used to get the water away is sloping down and has somewhere for the water to clear at the bottom (obvious, I know, but believe me I've seen the results when that doesn't happen). From then on you need to make every effort to keep the drain clear of silt. For a standard garden drain, I use S A P 20 as the aggregate, which is a scoria with aggregate sizes of 20mm. Scoria is ideal because it will never actually bind together (unlike builders' mix for instance) which means water will always be able to travel through it. Once the drain has started collecting water there needs to be a means for it to escape which is where the pipe comes in. Using perforated flexible pipe like Novaflo coil allows the water to travel away from the drain. Silt build-up can very quickly render a drain ineffective so to prevent this before it happens the whole drain is wrapped in filter fabric. I always cover the pipe in filter sock which is like a stocking that stops silt building up in the pipe. This is a cloth (now often sold as weed mat) which is placed directly into the drain on the bare soil.
Install the coil on a bed of scoria around 100mm deep, sloping downward so water runs off. Place more scoria over the coil, then fold the ends of the filter cloth over the top of the scoria.
Finally I usually use river boulders on top of the scoria as they look good in most gardening situations and keep the drain clear of litter.
Once you've controlled the water you are able to divert it around the lawn and away into a less-frequented part of the garden. You can dig a shallow sunken garden bed and plant plants that do well in water, like swamp iris. In a vege garden, raising the beds and taking a no-dig gardening approach will take care of heavy soil. Good landscape design is the best place to start; you'll be amazed how much water you can get rid of just through clever plant selection.