Personal opinion:
Intel > AMD
Nvidia > ATI
Errr... a very very simplified judgement....
To have a good verdict, one must judge on a case-by-case basis an not make such sweeping statements. And how can it be opinion when there are benchmarks out there? One merely has to compare the actual real world in-game performance and then compare the prices.
But in terms of raw performance, Nvidia currently holds the crown in all levels (entry, mid-range, high-end), especially with their fantastic 8800GT (the rise of the new 7900GS/X1950 Pro - cheap, affordable, and performs much better than its price indicates). AMD/ATI is just totally failing in the GPU department.
As for the Intel vs AMD CPU 'war', basically, if one will overclock, get the Intel models, but if one will not, then get the similarly priced AMD coutnerpart (and most of the times even cheaper, ex. AMD X2 6000+ vs. Intel counterpart e6600). But I overclock so of course I'll go with the core 2 duo.... nothing beats the feeling of having a CPU perform 65% faster than default.
Note that the amount of memory on a given video card (GPU - graphics processing unit) is not a good indicator of the overall power capabilities of the card. For example, a 256MB 7900GS is about 15x-20x faster than a 256MB 7300GS. One can see that even though the amount of RAM is the same, the performance is not.
As for the OP,
HP Media Center m7250n
2.8Ghz Intel Pentium D (A bit weak, Civ IV only uses one core really, so it performs only a tad bit better than a Pentium IV 3.0Ghz)
3GB RAM (MORE MORE MUCH MORE than enough, 2GB of RAM is excellent)
X300 Video Card (Weak, but Civ IV is not very demanding on the graphics)
I looked up this model and found out it has a PCI-e slot, which will enable it to have PCI-e based video cards. You said you have a $100-$150 dollar budget, I would recommend the following:
http://www.newegg.com/Product/Product.aspx?Item=N82E16814186013 - WOW, what a steal, a 7900GS for $99.99 after rebate. I strongly suggest you do not buy from any of the big-box stores (Best Buy, etc) because they often overcharge by charging double or triple the normal price. Newegg is the biggest US retailer of computer hardware and accessories and they offer great customer services so no worries if you are a bit iffy dealing with ordering online.
If you look around newegg, there are 8600GT models for sale also for about the same price, but the 8600GT is really underwhelming compared to their last generation counterparts... Just search for benchmarks comparing the 8600GT versus the 7900GS/X1950 Pro.
I wouldn't recommend to go to a more powerful model because your CPU would bottleneck it (moreso than it would already bottleneck a 7900GS); it would hold back a more powerful card from performing fully because the CPU would not be able to keep up with the GPU (Simplified - Basically, the CPU cannot handle the data the GPU is sending it fast enough which makes the GPU have to wait for the CPU to finish before the GPU can process new data).