I can't speak much to mapmyride, but I've recently started using cyclemeter. It was a bit daunting at first, for some reason. It's highly customizable and you can get screens to look however you like (I have a screen that shows my heart rate, cadence, av cadence, speed, av speed, distance, and time). As for accuracy, I don't know if that's so much to do with the app as it is a limitation of phone GPS chips, if I understand correctly. It seems that for really accurate speed/distance, a speed sensor on your bike would be the way to go. I use a Wahoo bluetooth sensor that feeds into cyclemeter, and a polar heart rate monitor as well.

Data from cyclemeter is exportable via email (gpx, kml, csv, and tcx formats) . I've never tried importing that data to other programs, but the capability appears to be there.