Probably the west side of the range, which has faster access to the beautiful high country. Pinedale, the nearest town to the west side of the Winds, is about an hour's scenic drive south of Jackson.
The best information source for those of moderate experience is Adkinson's "Hiking Wyoming's WInd River Range," which describes most of the major trails.
Most people for their first Winds trip go in at Elkhart Park, just above Pinedale, and hike in to Titcomb Basin. That's out of the grizz range, although you still should hang your food at night. Of course this route is (with good reason!) very popular. It also gets up to higher altitudes pretty fast, since the trailhead is at 9300 feet. I hope to do that one this August but will be doing an off-trail segment after leaving Titcomb Basin.
Be prepared for daily afternoon thunderstorms, wind, below freezing temps at night, possibly some snow (it will melt off fast, though). Or you could have beautiful weather (often with frost at night) the whole time. The same will be true of the Tetons or the GrosVentre Range if you go there.
When in July are you coming? Before the middle of the month, expect high stream fords and plenty of big snowbanks. Since (like the Pacific NW), Wyoming has been having a snowy, cold June, it may take longer for the snowmelt. It will be the height of the bug season--take a headnet and repellent!