I've done it twice via the Emmons Glacier and Camp Shurman from White River Campground. ~10,010 feet of gain. It is just a slog in the snow if noone falls in a crevasse and the weather is good. Once in July and once in August. Both times good weather. You walk around monster crevasses, and some you cross on snow bridges that don't look too safe, but we had no problems. The "trail" in the snow was good so route finding was easy. Start a little after midnight. Some had problems with altitude but all made it. I would not attempt it in bad weather and I am very experienced.
I would start hiking big hills as often as you can. I noted on the map that you can get in about 3000 feet of gain on Capitan Peak in New Mexico, near Capitan, NM. Looks like maybe 4 hours from Muleshoe. http://www.summitpost.org/capitan-peak/152409
Might get in more elevation on Sandia Peak by Albuquerque, but might be farther?
I'd do one or the other frequently. Maybe do Capitan Peak twice in a day. Maybe carry some water for weight and to douse yourself with if it's hot and dump some at the top to spare your knees on the way down. I've never been on the New Mexico peaks I am referring to so do your homework to discover the hazards to yourself, your car, etc and be careful on the long drives over there and back......long drives after climbing are very dangerous if you are tired. You can see hiking trails on this "my topo" website - I can't verify if they are right or wrong, or conditions, etc. http://www.mytopo.com/maps/index.cfm?search_string=capitan%2C+nm
If for some reason you can't do Rainier after you arrive, you can always run south to Mt. Adams a very nice climb - south side is non - technical just don't get off route onto the glaciers, or if you need to go farther south, you can go to South Sister near Bend, Oregon - a non-technical snow slog on the south side - very nice. Need to go farther than that to find good weather? Shasta in California has non-technical routes on the south side. An outstanding peak.