I would cut trenches and if you want to put drain tile into them that's fine...but leave them open until you get an idea of how they work. At least that is what I did. I got an area drained after several different ideas and tries, but in the process made every mistake possible.
After covering them, I have had a lot of problems with roots clogging mine. Mostly willow roots. Going to have to dig them up again. My advice is to make an access point every 75 feet for rooting them out.
Start now & budget for a roto rooter. We bought one... should have used heavier tile and bought a heavier rooter too. Like I said, every mistake.... we have about a quarter mile of creek frontage, so in that way the projects are similar.
Surprising how little slope is required on the drains. Get a slope gauge, a 2% slope is plenty. The trick is to make the slope so that the water can push any dirt through without leaving mud build up in the drain tile. Too much slope and the water outruns the dirt.
rScotty