The old MacBook Pro won't turn on. Is there some way to access the info on the drive before I recycle it?

Well, if it won't power on, the drive would not be accessible unless you performed surgery and removed the drive. Then you would need a bus-powered SATA III to USB-3 adapter to see if that drive would spin up as an external entity on another Mac.

What model exactly? It's pretty easy to take out a SATA drive, but it gets more interesting when it's a newer MBP. The various types of SSDs that Apple has used are confusing to figure out, and there are limited interfaces that work.

You can get the exact model of the laptop by entering the system serial number here:

Check Your Service and Support Coverage - Apple Support

Basically any non-USB-C model Mac can have the internal drive removed and connected to another Mac with a USB adapter or enclosure. If you have a USB-C Mac, then your options will be extremely limited as there are only a couple models where there may be a small chance of accessing the data some of which can only be done by an AASP or professional data recovery service.

You should always have frequent and regular backups of your computer and all external media (including the cloud) which contains important & unique data.