Proper grading away from the house is the single most important after construction component of keeping a house dry. If your downspouts aren't draining water away, your grading is likely the culprit. That is not DIYable.

I worked on a house with similar problems where the solution was to install buried conduit(they used PVC Schedule 40 pipe) at a slight downward angle(like sewer piping) terminating in an area away from the house. In this case that area was the street gutter. Since this area freezes in winter, a freeze joint was used. A freeze joint is simply a gap in the piping which allows water to escape to the outside if the underground pipe freezes.

One corner of our house has a downspout which is connected to an underground pipe which is U shaped, the termination is about 10 feet away on a down hill section of the yard. The effect is exactly the same as a long down spout extension, but is invisible.

The first solution was done by a crew with a trencher, but the second could be done with a shovel, since the depth of the underground section is only a foot or so deep.

I included a Google page on the subject.Google downspout extensions

I put the downspouts into solid PVC/ABS piping that eventually meets up with perforated piping and terminates in popups far out in the yard. However, these don't always drop down after use, so remember that when the yard is mowed!

I've also used the flexible drainage pipes--much cheaper so they're preferred by contractors--but found that they clog up in a few years with roots while the solid piping can be snaked out. For easy access, include a "Y" fitting where the downspout enters the underground pipes.

Note: Some municipalities absolutely prohibit you from dumping your roof water into the street or on sidewalks.

It's also possible to integrate the drainage with stone lined depressions into garden areas.

The only downside I've found was that grass above the perforated pipes stood out for its well-watered look!

Well, rainwater harvesting may be a way to consider:

Laser levels are very cheap, and good, nowadays (I think my first one was several thousand dollars). I remember when the Robo Laser came out for $350, and it was devestating. :-) That being said, find out your level, and then start figuring out what's lower, where you need to drain to. If you need help with that; ask us. There are at least two dozen builders and architects who post on these forums who will help.

Thank you everyone for your feedback! We actually have excavating professionals in the family, so it is kind of DIY. We do have the grade sloped correctly, but with heavy rains lately there is erosion of the new fill and it makes it difficult to keep grade perfect. We are hoping to get the gutters buried and draining out away from the house and then restore proper grade. Hopefully that makes sense.

I am leaning towards something like this hooked up to a PVC pipe that goes to a pop of drain in the yard. We are in a cold climate, so I am worried about water freezing back up into the gutter if we connect the extension directly. Thoughts?