Did I hear my name?(How's it going Angel

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The best place to start, IMHO, is with a low end application, possible one of the many MetaCreation offerings. These will help you learn the basics of 3D modeling and also give you a better grasp on what you want to do with 3D. You mention animation which I won't try to talk you out of in any way, but you may decide that you like focusing on model creation or stills more, whichever way you decide you want to go starting with a low end, which tends to mean lower cost, application will help give you a basis to decide which way you want to go(higher end apps tend to be aimed more towards a particular audience).
When deciding on which application to get, in the real world cost is a major factor. You mention that $200 is a bit high, and in all honesty many plug ins cost quite a bit more then that. In the sub $200 price range you are looking mainly at Simply3D type applications if you want the latest version, but going with older versions of more powerful software may be a better bet for you(that is how I almost always buy things, even if I want the latest it tends to be much cheaper to buy the older version and pay the upgrade price).
A good way to go about this might be to have someon with an e*ay account place a bid for some older software for you. I just ran through a few searches and found Infini-D 3.1 going for less then $20, that was a $500-$600 application when new if memory serves. And if I recall correctly, Infini-D was one of the better, if not the best, low end 3D solution available at the time. Also available right now is version 4.5, but that is in the ~$200 range.
Another option could be RayDream Studio, another popular application, also Bryce, Canoma and Poser come to mind. These may not be as good as some of the less popular alternatives, but odds are that you will be able to find more help for any problems that you run into using one of these.
As far as how much you can do with them, I think you will find that you are able to do quite a bit, and it will take you some time before you start complaining about missing features. Up until not too long ago I was happy overall with Caligari's trueSpace for my uses, even though I had used significantly more powerful software, I just didn't see the advantages(at least not enough to warrant the costs), and I've been doing this for many years now.
Talent is not required, you can teach yourself to become quite good without any natural talent, though it does help quite a bit(
Excaliber2013 has some amazing talent, a he!! of a lot more then me for d@mn sure). You mention that you alread do well in art, the next step is to learn how to work in 3D. For this, it is much easier to start with an application that deals with the more basic elements. Learning something like Maya, SoftImage or even 3DStudioMax is difficult enough for someone who has been working with 3D for years, let alone someone new to the field.
Computer power- You don't have enough, but neither do I

3D visualization is one area that "fast enough" does not exist. Will your system be able to do it? Yes. Will you be happy with the speed? Highly unlikely. You will learn that "good enough" and "a lot better" are normal terms, fast enough is a pipe dream

I have three different machines I'm using right now, building a fourth, and three out of those will be used mainly for rendering and nothing else(LAN). The slowest of those is a PII 266 which is plenty "fast enough" for the average computer user, a complete dog for 3D vis. The suggestion for a GeForce is a very good one, it made my system a
*lot* faster working with models, particularly ones with high levels of geometric complexity. Fast enough? Of course not, but it allows me to work with higher levels of geometry at acceptable speeds, a five to ten fold increase over my TNT with Athlon 550, no joke.
Sorry for getting long winded, in short I would check out e*ay or another auction site and look for older, used versions of software to get a decent package at a decent price point. $1000 in 3D is a drop in the bucket.