Ok first off the background you gave us is very helpful. Knowing what the computer is for helps so you know what you need to get the job done. Your title says you have $1000 budget but I'm guessing you would prefer to spend a little less if possible ($800).
CPUs:
Intel CPUs come a few flavors. The main ones are: Core i3, Core i5, and Core i7. Generally speaking gamers get one of the top end Core i5 processors. They lose hyper-threading that the Core i7's have but no games really take advantage of hyper-threading so its not a big deal. Hyper-threading is basically adding virtual cores to the processor. So a quad core i7 CPU will have 4 physical cores and 4 virtual ones (the hyperthreading). AMD does something similar with their Oct core CPUs such as the 8350. It has 4 physical cores and 4 virtual ones. Virtual cores are "weaker" then physical cores. Games have to be programmed to take advantage of virtual cores and none do that as this time.
AMD CPUs also come in many flavors. Vishera is the latest generation of AMD's CPUs. Overall AMDs highest end processors do not compete well with Intel's highest end but then you're comparing a $200 CPU to a $1000 one. They do compete well with the Intel Core i5's and entry level i7's; but it does depend on the task. Intel is tends to do better in single threaded applications, such as many games. In multi threaded applications AMD starts to shine. Newer games however are being written to be multi threaded.
Which one should you get? Really, its up to you. A high end Vishera like the 8350 is a solid choice. A few good Intel CPUs would be the 3570, 3450P (the Intel built in graphics card is disabled, that's what the letter "P" stands for) Since you're not interested in over clocking just make sure to avoid any Intel CPU that has the letter "k" at the end of it. You're paying for the CPU to be unlocked for more over clocking. The 3470 you mentioned is fine as well.
Motherboards:
I'm not an expert on motherboards but a few things to do when shopping for one:
Select your processor first, then choose your board. If you go Intel make sure the board is listed as an Intel board. Same goes with AMD. Then you need to check what Socket your CPU is made for. If its an Intel 3470 then you will need an LGA 1155 socket Intel Motherboard. If you go with the Vishera 8350 then you will need an AM3+ AMD motherboard. Without going into every single detail I would buy a motherboard in the $100 price range. Going above $150 tends to get you into boards that are simply better at overclocking. A few more things to look at though: how many hard drives/SSDs/optical drives do you want? Make sure there are enough SATA ports on the motherboard for those. Also, Try and get at least 2 SATA III ports on the board. SATA III is faster then SATA II, etc. Newer SSDs and HDDs take advantage of SATA III's extra speed. Does the board have USB 3.0? Since you're building a new computer you might as well have it. No point in dating yourself right from the get go. A number of good motherboard makers: Asus, AsRock, Gigabyte, MSI are all popular ones.
Graphics cards:
AMD verus Nvidia... really don't want to go here but both companies make good products. Each one slips now and then too. Right now AMD has the price/performance ratio advantage. Their cards tend to be more cost effective. I'd recommend getting a HD 7950. That may cost more then your CPU but most games are more GPU (Graphics Processor Unit) intensive then CPU intensive. You said you don't want to use multiple GPUs so you're better off getting a higher end one to start anyway.
Power Supply Unit:
I like Antec and Corsair. Both are reliable and good quality companies when it comes to PSUs. For a single GPU setup you shouldn't need more then a 650 watt unit. You'll notice they come in different certifications such as 80 plus, 80 plus bronze, 80 plus silver, 80 plus gold, and 80 plus platinum. That tells you they are certified to be X efficient with power. A PSU is taking electricity from your wall outlet and converting it to a different form for your computer. During that conversion it loses some of the electricity, like a car does converting gas into kinetic energy. I don't know where you live but in the USA electricity is cheap so 80 plus bronze is all you should shoot for. Other then that go with one of the two brands I listed and make sure its 600-650 watts and you'll have enough power to power your computer and the PSU's will be made from quality parts so you're less likely to have it damage any of your parts. Not to mention it will be more reliable.
Solid State Drives and Hard Drives:
SSDs are not very cost effective, there I said it. I know some will disagree, but they're not. They're fast, sure but you pay a lot for it. Most people only use a SSD for a boot drive anyway to load Windows. I think that's kind of silly, since you're paying a lot more to load Windows faster but then what? Load all your programs on a 1 or 2 TB hard drive and you're back to normal loading speeds. I recommend you either buy a SSD that's at least 300-400 GB and load Windows and all your applications on it, then have your data (pics, videos, music) on a hard drive or just put everything on the hard drive to begin with. Just my advice. Anyway, this is where the SATA III port on the motherboard will come in handy. Regardless if you go SSD or HDD you will want a storage device that uses the SATA III port rather then SATA II. There are really only two hard drive brands anymore. Seagate and Western Digital. I personally like the Western Digital 1 or 2 TB Black (Blue is for economical and Black is for performance) drives. If you go SSD then I recommend Corsair and/or Samsung. I know a lot of people like the Samsung 840 series, they have really good reviews.
I have to get going now, but I hope that helps you and I wish you luck!