1. If you have an ethernet port on both of the computers, you can network the two by using an ethernet cable and transfer the data from one to the other. But I doubt it that you have an ethernet card on your old computer.
Ethernet port looks like a phone jack except a little bigger.
2. Installing the external hard drive!?
Your simply going to hook it up to the USB port and probably just install the driver that comes with it (a cd). Then you can just copy and past or drag and drop the items into the new drive (that'll be shown in 'my computer').
Usually there are great deals on internal hard drives and kits that make them externalLike this one for example, mostly because of rebates the package becomes cheaper than external ones.
So, you after you are finished you can just install the internal drive into the new computer and have two hard drives and when needed, just take it out and put it into the enclosure.
Or you can just buy the enclosure and put the hard drive from your old computer into it and hooked it up (USB) to the new computer and transfer all the data that you want from it into the new computer.
3. Just buy this and transfer data couple of times. Pictures are usually 3MB or less, so lets say if they are 3MB each, 256 MB stick will hold 85, and you just transfer those 85 over and delete them from the stick and transfer another 85 over again.