The 14ers in Colorado do look like the closest ones to you.
You might be able to do Tokewanna (UT, 13k) in a 2 day weekend, but it could be rough on your knees. 3 days would be better for getting started.
http://www.summitpost.org/tokewanna-peak/152823
LeadvilleNative's circuit sounds like a big test, but not really what you are doing on Everest because of the heavy pack. If you want to try it in the winter you will need time to develop a network of mountaineering partners. Going solo on that without much experience wouldn't be good.
Just getting some time hiking at 14k in Colorado is the main thing. Assuming you can go up and downhill fine in the 9k-13k range.
Aconcagua before Rainier is OK as long as you test yourself well at 14k first. Aconcagua is pretty high, at 23k. It sounds like you are trying to rush it. I'd do Rainier first.
Everest is weird now because of the $ and the fixed ropes. You are basically hiking up a series of footsteps with an ascender on the fixed rope. Little or no pack; most stuff is carried for you. It requires $, the ability to function at altitude and some endurance.
Conventional mountaineering is done mostly unroped because of the long distances and time it would take to anchor the rope to belay. So you will be in some situations where if you slip, you probably die. Different from having a fixed rope. You have to develop skills for getting up terrain in careful control over many hours. Both this and Everest require moving efficiently.