If you are planning on spending that kind of money i would use the mobo+processor bundle that linux009 posted. I wouldn't use the one that Araneop put because your system will be unbalanced. It won't hurt it at all to use it but you won't see any better or worse performance with it, you will just be spending more money.
The memory that comes with that system isn't very "modern" either, i would recomend getting some more of that, because you only had 1GB before i'm guessing that you have a 32 bit operating system. Because of this I would advise you to get 2GB of memory too. You can get this very cheaply, $20-$60 depending on quality. A $20-30 would be sufficient for you though as you aren't gaming or using it for anything greatly strenuous.
http://www.newegg.com/Product/Product.aspx?Item=N82E16820145590
http://www.newegg.com/Product/Product.aspx?Item=N82E16820227124
http://www.newegg.com/Product/Product.aspx?Item=N82E16820148204
They are all good RAM, will give similar performance, you won't notice much or any difference, just giving you a bit of choice.
As for your question on whether the motherboard is good:
Asus are well known for making high quality and stable motherboards. I am using one now, that, sadly, i am going to have to get rid of fairly soon because i've just about reached capacity for it and can't upgrade on it much more. It has given me no problems at all over the past 2 and a bit years i've had it and i've gone through alot of hardware in that time, all of it it has coped with fine. There have been a few hiccups but usually it has been the new hardware fault, not the motherboard. I would deffinately recomend getting the board posted.
As for the difference between dual and quad core processors, there is alot of difference, but not noticable for your needs.
If you are multitasking (gaming, browsing, listening to music all at once), quad would be your friend. Gaming in general preety much NEEDS quad core now for high settings. General usage, ie browsing the internet, word processing, e-mailing, viewing/sorting pictures etc and some gaming, dual is fine. They have similar power usage, even though quad core processors have more power so there is no difference there,