I am looking to built a cheap computer mostly for web surfing and basic tasks. I want it to be fast though. My current computer is starting to take forever on startup and has trouble handling websites with a lot of multimedia, and I have a very impatient husband who is frustrated with it. He goes to a lot of websites like Gamespot and IGN to watch videos and things, and there's usually so much going on that the pages take forever to load or freeze and won't do what he wants them to do. I need a new cpu, mobo, ram and gpu. I have everything else, although my power supply is only 350W and about 4 years old, so that may need to be replaced too. I don't care about AMD or Intel, although it seems like AMD might be better on a budget. I don't want to mess around with overclocking. I don't planning on using it for gaming, but if it's not too much, I wouldn't mind having a system that's capable of playing games decently. I don't really have a set budget, but I'm hoping somewhere around $300, probably no more than $500.
You really don't need more than the integrated video for your intended use. The upgrade path might turn out to be a pain depending on what your computer is. Is is an OEM like HP, Dell, etc...? If it is, they often use proprietary components that make it a real pain to upgrade. You also probably won't be able to re-use your Windows installation. You certainly can't just take the HD out of your old computer, install it in the new computer, and expect it to work.
For your purposes, I would get a motherboard with integrated video and skip the video card. And since you are skipping the video card, I bet your old 350W power supply will work just fine. You can always get a new power supply and video card later if the integrated solution doesn't suit you.
The current computer was a custom build done by a friend awhile back, and I've added a new hard drive and optical drive, plus purchased an os in the years since. So it seems like such a waste to replace it all, when many parts are still just fine. It's in an old Antec case with Antec power supply, I'm not sure what model. Will pretty much any power supply fit the case? Or do I need to take measurements or buy another Antec? I've never build a whole new system, just replaced parts here and there, so I've never messed with the power supply.
I went with the ASUS mobo over the Gigabyte one that was suggested because it looks like there is more space for the video card, and the one I chose has a large cooler and takes up two spaces. I'm basing this off the pictures of the boards, so I could be wrong. I'm not really sure which would be better, they look very similar to me. The video card is probably more that I need, but it seems like a very good deal. Same with the power supply.