Build new pc or upgrade current one???

You need to start fresh. That old computer won't really have anything you can save besides the HDDs IF they are SATA not IDE. You might could sell the rest for $50-$75

You definitely need to upgrade, but you can salvage a DVD drive and any SATA hard drives.

PCPartPicker part list / Price breakdown by merchant / Benchmarks

CPU: Intel Core i3-4130 3.4GHz Dual-Core Processor ($119.98 @ OutletPC)
Motherboard: Gigabyte GA-H87-D3H ATX LGA1150 Motherboard ($99.99 @ Amazon)
Memory: Crucial Ballistix Elite 8GB (2 x 4GB) DDR3-1600 Memory ($74.99 @ Amazon)
Storage: Samsung 840 EVO 120GB 2.5" Solid State Disk ($99.92 @ OutletPC)
Video Card: Asus GeForce GTX 650 Ti 1GB Video Card ($134.98 @ SuperBiiz)
Case: Antec One ATX Mid Tower Case ($54.99 @ Newegg)
Power Supply: XFX 550W 80+ Bronze Certified ATX Power Supply ($49.99 @ NCIX US)
Operating System: Microsoft Windows 8.1 - OEM (64-bit) ($99.98 @ OutletPC)
Total: $734.82
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2014-01-09 10:52 EST-0500)

Here is a good option for you, should do well with most task, gives you a fast SSD for your OS and apps, and will be a big upgrade over your current computer.

If you have any plans for overclocking your CPU, you will need to switch to AMD because the i3's are not able to do so. Unless you have plans for CPU intensive activities or high-end gaming though, the stock speeds will work fine.

Please remove your personal email address, and never ever ever ever EVER post it on a public forum. That is by far the best way to get enough junk mail to basically necessitate getting a new email address.

To answer your question: It all depends on what you are doing with your computer.

If you are just using it for multimedia, office, and web browsing then (depending on your CPU) you may be able to get away with a few simple upgrades. Note that this is assuming that you are running a 2.5GHz Core2Duo or better; Get a SSD for your system drive, and a nice new big HDD for your data to replace all of your old drives. This will get you much better throughput on your drives, and clean up the unholy organizational mess of running 3 HDDs. As drives are the largest bottleneck of almost any system, this is going to be your single biggest upgrade that you can make. Next, make sure that you have at least 4GB of DDR2 800 or better memory, and if your system cannot handle DDR2 800 then consider upgrading the entire thing. Lastly, upgrade to a newer basic GPU to help offload those resources. In my office PC I am using a GeForce GT 610 which is more than good enough for most tasks and running multiple monitors... but if playing some games then look for something a little bit better. Go fanless on the GPU if possible.

Now, if you are running something slower than a DDR2 800 capable 2.5GHz Core 2 Duo (or AMD equivalent), or your are going to be using your computer for something other than basic office, web, and multimedia (such as games and production software) then it is time to upgrade to a new system. If your budget is under $500 ten it is often best to go with a pre-built Dell, HP, ASUS or other such system. They can simply make things smaller, quieter, lower power, and cheaper than you can for a cheap general use PC, while offering a decent warranty on the system.
If a prebuilt system is not for you, then consider something like a NUC (which is what my wife will get on her next upgrade) or BRIX, or some other similar small-formfactor system that just needs an SSD and Ram added to it to make it work. Then pair it with a NAS for bulk storage, and an external DVD burner or BluRay drive if you really still need a CD drive of some sort. These systems are small, quiet/silent, you can mount them to the back of your monitor, they use extremely little power, and there are very few parts that can break in them.

If looking at something larger or more ambitious then look around the forums for parts suggestions. Depending on what exactly you want to do then there are just too many variables to account for.

As for selling old parts, if the system works you could probably sell the entire system (with a functional OS) for $50-100USD.... not likely worth the effort, or worth getting rid of a perfectly good copy of Windows. It would make more sense to strip out the HDDs and donate it to a local nonprofit refurbisher if you have one in your area and take the tax write off.

You could get by for a little longer, but that CPU will be a bottleneck for today's GPUs. I personally would start fresh.