What we're all assuming is that this film transfer facility is Mac-based, or at least Mac-friendly.
If they must give you the files in AVI, you will need to convert them (a lossy process) to another file format. While there are several ways to do this, it's time- and disk space-consuming.
If they can convert your old movies to a digital camcorder format (e.g., Mini DV, AVCHD, etc.), then you can import the files/media directly into FCPX.
When my wedding was videotaped, they did such a horrible job of editing the footage, I asked for the raw footage on DVD-ROMs. (It was so bad that my wife burst into tears upon seeing it -- not tears of joy, either.) Then I imported those files into Final Cut Express and make a good wedding video. You should be able to do the same thing with digital camcorder footage from the film transfer house.