Can someone please give me some guidance on the best way to tie my gutter downspouts into some underground drainage so the water will not just pour down my yard on top of the lawn? Also want to make sure the pipes avoid getting clogged over time to avoid digging it all up again.

Although I haven't researched the entire design process, I can tell you what I know:

1) A catch basin under the downspout will catch leaves before they reach the pipes. This is basically a grate with a basin lower than the discharge.

2) You have to get the water away from the house with non-perforated pipe.

3) You can then use perforated pipe into a leach field. Forget which way the holes have to point. The pipe is covered with a fabric to prevent sediment from clogging the pipe. There needs to be a particular kind of gravel around the pipe. There should be a slope. Not sure how much.

4) The easiest way of doing this is by using corrogated pipe.

5) Ideally a drainage container can be used to collect large amounts of water underground. It can be combined with the pipe with the holes if necessary.

6) You should be able to find what the average rainfall per storm is fromand from your roof footprint and use that to design the system.
If it's 1//2" of rain, fint the horizontal footprint of the area served by the gutter and convert volume to gallons.

7) The depth, I have never been able to determine. It does not appear that under the frost line is necessary.

8) Home Depot has a nice selection of components.

9) I'd like to do this with rigid PVC, but in order to do have the catch basin, a small amount of flex pipe will be necessary.