8:02 p.m. on April 17, 2012 (EDT)
Beckey.
That's me on the left, Eddie Joe middle, Beckey on the right. Last Dec 8.

Fred just keeps climbing, and climbing, and climbing.
You got some good suggestions up above, and (typical of the Internet) some not so good.
Step at a time is not just good, but the only way.
You said your parameters are:
Thinking more along the lines of over the next 10 years climbing some continental US mountains, rockies 14k climbs, Mt. Ranier, maybe Kilmanjaro, and the ultimate goal would probably be Denali, or something in the Himalaya. I understand that this is a process and with a family with 3 young kids and living in the SouthEast this may take some serious time with only 1 major expedition every year or two.
Don't believe the SE lacks climbing. There is great technical rock climbing there. Specifically, there is excellent rock in NC, GA, TN, AL, AR, WV, VA, all not that far from SC.
The 3 specific peaks you name (Rainier, Kilimanjaro, and Denali) are all pretty much long slogs with no real climbing by the most popular routes. Rainier and Denali involve snow and ice and glacier travel. Kili is a long trail in pretty reasonable shape by any of the routes to the summit (there are climbing routes on Kili, but those are not part of the trails to the summit). Any hiker in reasonable shape can do Kili, IF (and that's a huge IF) you take it slowly ("polepole" as the local guides you are required to use, under Tanzanian law, will continuously tell you). I have a trip report on Kili here. I would point you to my trip reports on Denali, except that my ISP decided to eliminate their "free" home pages. So that went away, along with TRs on the Mexican Volcanoes, Antarctica, Europe, etc etc. Oh, wait! I have a TR for the 40th Anniversary of the first ascent of Mt Vinson in Antarctica here! That might be a bit on the expensive side for you, though.
Most of the Rockies and Sierra 14ers can be climbed by no more than 2nd class routes, though some have 3rd or 4th class as the easiest routes. If you set off to do all the State High Points, the hardest are 4th class. Except Denali, which is a serious glacier climb, though mostly a long snow slog at high altitude, with only a couple of sections with real climbing, and those have fixed ropes (unless you decide you want to do something like Fathers and Sons - but your itinerary sounds like you are wanting to do peak bagging, not technical climbing).
Given your itinerary, I would suggest you look into a couple of guide services that run climbing schools with a mountaineering slant. This is the fastest way to build your skill level safely, though it will require traveling across the continent for a couple weeks over 2 or 3 years. Going by friends who are professional guides and in the business of training climbers with a similar itinerary, I would suggest looking at American Alpine Institute (very much aimed toward training climbers looking to do mountains like Denali - based in Bellingham, WA) or Alpine Ascents International (based in Seattle). Both AAI's conduct a lot of their courses in the Cascades, notably on Mt Baker and Mt Rainier. You get some altitude experience in the Cascades on these mountains.
Exum Guide Service (based in the Tetons) spends more time on technical rock. If you had expressed interest in technical rock, I would have recommended Yosemite Climbing School (in Yosemite National Park) or Mountain Adventure Seminars (based in Bear Valley, CA - they run glacier classes on Mt Shasta, a place to experience some of the fiercest weather in the lower 48 states and excellent location to develop skills needed for climbs like Denali - Shasta is deceptive, with people just strolling up Avalanche Gulch in good weather, yet a large number of injuries and deaths among the unprepared and unskilled when the sudden fierce storms hit).
IME, located in New Hampshire at the base of Mt Washington (North Conway, NH) runs excellent courses in technical rock and technical ice. Though, again, you did not mention technical ice in your parameters.
There are many other climbing schools. Important thing to look for is location that matches your goals. Which is why I mention the AAI's - snow and ice, glaciated peaks, and altitude that the Cascades offer, and both run courses in the Alaska and St Elias Ranges as Denali prep and on Denali itself.
Another place to get altitude experience, though not difficult glacier travel, is the Mexican Volcanoes - Ixta, Popo (may or may not be open, due to volcanic activity), Orizaba, and several smaller, but interesting ones.
Consider including the Andes in your itinerary. Yeah, the Himalaya are the glamor mountains. But climbing there is expensive compared to almost anywhere else. Plus there is still a lot of political turmoil. Tourists (including climbers) are usually safe enough politically, but not always.
By the Andes, I mean the equatorial Andes in Bolivia, Ecuador, and especially Peru. Aconcagua, which is effectively on the Chile/Argentina border, is popular, which means often crowded. The normal route is a none-too-pleasant slog, but more of a long hike than a climb. Make no mistake, though, the weather can turn ugly (it is high altitude, after all, and not far from the Pacific and incoming weather). The Cordillera Blanca, in Peru, has a plethora of gorgeous mountains, and at the right time of year, very pleasant climbing weather.
Hmmm ... you said you are 37. Yup, that's about the right age for YMIS to start tailing off. Well, never fear. If you play your cards right, you will still be close to your peak when you get to the age that I am and probably to the age that Fred is.