My computer started intermittent freezing and became really slow. Restarting fixed the problem, but on one occasion it didn't and I reset the computer while the system was starting.

Then I got a "disk read error. press alt+delete to reset"

I've recently re-installed windows 7 on another ssd drive. However, I have important information I need from my old drive. My bios can detect it in the boot menu, but when I choose that drive as the boot drive, it reverts to the new ssd drive. I can also detect it in device manager, but I cannot access it.

Is this a file system problem and is there a way to repair or restore the drive without losing the data?

when computer start showing symptoms, you mentioned, this indicate, that something is wrong with drive.
If you are not able to access this drive, when boot from SSD, you can try with linux boot from CD, USB, or hirens.
If you are sill not able to access data, then stop working and contact professional data recovery near you.