You have several options.
1. You get him to build the computer for you, with the new (hard drive) HDD and ask him to put the old one in as well. This will then show up as an extra drive in My Computer and you can simply copy the data across.
2a. You can buy a new computer, take the HDD out of the old computer and put it in an 'USB IDE enclosure' - This will basically make it an external hard drive. You then plug it in to a USB port on your new computer and copy the files across as above.
2b. You can buy a new computer, take the HDD out of the old computer and put it inside and do the final sentence of option 1.
3. You could build it yourself, (jperhaps even getting the parts he suggested) save a couple of hundred on labour costs and when assembling it put the old drive in with the new and copy the files across.
The best thing about #3 is you get to learn a lot about computers, and with the internet, it really isn't that complicated any more! I've built a few machines now myself, and really enjoy doing it.