It sounds like you already had an external drive. Did it not have a recent backup on it? Preferably a cloned backup. The new machine should be able to use that backup to get your new one setup, especially if it was a clone. Migration Assistant usually comes up almost as soon as you start a new Mac.
Buy a Buffalo, if you want, but you'll not know what's inside the box which, IMHO, is more important than what the box looks like. All you will know is the size and the interface that you'll use to connect to it. For close to the same price (and probably less) you should be able to buy the drive and an enclosure and get the best drive at the same time (minimum 3-YR warranty, more is better, of course). You will need to know how to use a screwdriver, however. A small phillips head, usually. If there are more than 6 screws, it will be unusual (four hold the drive to the frame, the other two hold the enclosure together). Cable should come with the enclosure, of course. Get one that is at least twice the size of the internal drive in the new Mac, BTW.
First thing to do with any new drive—format it, even if it says it already is. There may be some useless utilities on it that probably aren't worth the space they take up, anyway.
Second thing to do is set up a backup plan. The new Mac will be ready to use Time Machine, of course. You've paid for that app, might as well use it.
Third thing to do is update as many apps as possible, many may not have Lion capable versions, of course, if they need Rosetta.
Fourth thing will be to see how much of the old stuff you can get off the old machine. I'm suggesting that as the last step as it sounds like it is not too stable and you'll have enough to do without doing something frustrating or possible causing corruption on the new machine.
It might be worthwhile to take the drive out of the G5 and put it in the new enclosure to see if it works better. You might even find a $20 Seagate 'adaptor' at Best Buy that will let you simply plug in the old drive! I've been using one for over a month with not a single problem. All SATA drives simply plug into the adaptor because all drives have the I/O and power plugs in exactly the same place. They are
impossible to plug in wrong, unless you have a very big sledge hammer and someone willing to hold the drive while you pound on it.
It can come in handy to have another "enclosure" laying around to test a drive that you think is failing. And you can't beat $20!