Yes your SATA III HDD will work in the SATA I port, just at the SATA I 150 MB/s speed. Some of the older SATA II cards had a jumper to set for the slower speed but I doubt that your newer HDD has to be manually set on the back.
Later, when you upgrade to a faster MB with SATA III ports, your HDD will run at native speed, without a handicap!
SATA III drives are backwards compatible with SATA I ports. The device will run on SATA I speeds and not the designed SATA III speeds.
Your bigger concern will be with the capacity concerns, as older SATA I motherboards may not support drives over 1TB. I would double check to make sure your motherboard is capable of handling a larger drive (if that was your plan).
If you are looking at using a SSD, I would only recommend doing so if your motherboard supports AHCI in the BIOS. It is a benefit to the performance of a SSD to run in AHCI (though not a "MUST" however).