July 4th is late for Hood, rock fall can be a big issue. I would look at Adams Glacier on Mt Adams, Coleman Demming, Coleman Headwall, North Ridge, Easton, Boulder Glacier routes on Mt Baker or North Face, Price Glacier, Fisher Chimneys, Sulphide Glacier on Mt Shuksan.

It usually rains over the July 4th weekend so I would have alternate plans on the east side of the range e.g. Washington Pass (Liberty Bell, North and South Early Winters Spires, the Wine Spires, etc) the Enchantments/Stuart Range, Mt Maude.

ExcitableBoy makes some good suggestions, especially if you are looking for glacier experience prior to Rainier.
On the other hand, if you just want a day climb close to St. Helens, do the standard route on Adams. If you're in good shape, it's a day climb. I've done that route a bunch of time, with skis, with my 13yo daughter, etc. and never spent the night on it. (Note that although they are only ~40 miles or so apart, the drive from Adams to St. Helens takes *forever*).

Hey Fletch,

What route are you planning on for Rainier?