Congrats on your purchase! The 627 is one sweet revolver. I would test various inexpensive rounds to find one that works well for plinking, etc. Then try some defensive rounds to see what works in that department. You of course have the option of going .38 or .357. When you found them, buy a handful of boxes and enjoy your revolver. One defensive load that I can recommend via experience is the Federal 110gr Hydra-shok loading.
I took it out to the range and put 100 rounds through it and had a great time. It is rock solid and was just a little low at seven yards. Three clicks of up on the sight and it was dialed in with .38 special 130 grain FMJ's. The rubber grip feels great and fits my hand just right. I could not be happier with it!