Firstly, Welcome to Mossberg Owners! As youre able, stop by the new member area and say hi to everyone!!!
By all means, shoot as much as youre able. Quite a few of us have been testing A BUNCH of different ammo, primarily 00 buck.
The Winchester ammo, while not my favorite, is widely used and your 590A1 will handle it with the greatest of ease!
Have you ever fired full powered loads through a shotgun? You mention being new to the shotgun platform, but I dont mean to assume you have never fired one. Reason I ask, there a few things you can do to lessen the blow and impact on your shoulder and cheek weld from high brass/higher velocity shells.
When you get a chance, check out the reviews and watch videos we have posted on different makes and types of ammo and then get out and shoot for yourself to decide which will best meet your needs. Also as Dan mentioned, let us know what goals you have in mind for your ammo, and we'll help as much as we're able!