Don't be put off by the low elevation of the mountains in this area. "Big" mountains in the Cascades have a much lower elevation than places in Colorado, for example. A 14,000 foot mountain in Colorado has a base of 10,000 feet, so only 4,000 feet of relief. A 7,700 foot mountain like Ingalls has a starting point of 4,200 feet, for about 3,500 feet of relief. As Excitable Boy has suggested, if you had good weather and wanted to make an attempt on a worthier goal, Mt. Stuart (right next to Ingalls) at 9,400 feet has a vertical mile of relief, more than any 14'er in Colorado, and it's a "big" mountain. A really "big" mountain.
If you really had your heart set on Mt. Rainier and weren't too concerned about getting to the top, many people like to make the trip up to Camp Muir at 10,188 feet. This is the over night camping area on the busiest route on Mt. Rainier, and where the "climbing" part really begins. The trip from the parking lot at the Paradise Ranger Station (5,400 feet) means that you'll be gaining 4,800 feet to Camp Muir over the Muir Snowfield, a permanent snowfield. You'll likely be traveling through some fresh, unconsolidated snow also, as this week, the weather is starting to change to a wetter, cooler pattern, so you may need snowshoes or skis to go up it, but you won't need to rope up. After Muir, it's all real glaciers and roped climbing. And just because you won't need to rope up on the way to Muir, don't take this trip lightly. The weather and visibility can get bad in a real hurry. I've been up there in bad weather, and you can't see a thing. The natural fall line down the Muir Snowfield takes you down to the Nisqually Glacier, not back to the Paradise Ranger Station, so you need to be prepared with a compass or GPS. A lot of people have gotten lost in bad weather going down that snowfield, and more than a few have died.