If you haven't started watching either show yet, you can watch Grey's first, because Grey's has a strong plot connection between each episode, which makes it more informative to watch, whereas House is one story per episode, and you need to stick with it and be patient to find out what's interesting.
Grey did start out excellent and was a ratings winner locally in the US, but there was a period in the middle where the plot was obviously dragging and boring, and it took perseverance to stick with it, but the episodes so far are still excellent, and there's obviously more of it in there as well, with good looking men and women. But in fact, Gray can not exactly be called a medical drama, just takes place in the hospital inside the serial drama, will be the characters' feelings mostly, after all, the story has developed for six years, the characters are inevitably divided between them, so the plot does not have to be too serious about the plot.
House is one of my favorite American dramas at the moment, chased for many years, the characters inside are very emotional, if you insist on watching it, I believe you will fall in love with Uncle House. It's now in its sixth year as well, and the writers have put a lot of heart into the new season, and the plot is very interesting. If you're a medical major, you can also learn a lot of useful knowledge, if you're not a medical major, you may have a headache with a lot of technical terms in it, but it won't affect your understanding of the plot at all haha.
Also there are some medical dramas are good, such as "ER", Uncle Hauser in "Dr. Hauser" inside often watch this drama, is the handsome George Clooney when he was young, classic drama.
And the first few seasons of "The Plastic Surgery Room" are also good, the make-up in it is simply amazing, so realistic that it's scary, it won the best make-up award, you can watch it.
These are my true feelings, I hope you pick your favorite American show.