The speedometer is mechanical, driven by a cable with a squared off end that goes into the speedometer gauge and a corresponding end that goes into a gear on the side of the transmission. Frequently the cable is the problem. It hasn't been lubed in forever.
Disconnect it from both the speedometer and the transmission, then turn one end (if you can) and see if the other end turns. If it does, you should lube the cable, reconnect everything and see if it works.
If you can't turn it, try pulling it out of the cable housing. If it comes out in one piece, it's at least intact. Inspect it for kinks and frayed areas, and look at the both ends carefully. If their edges are rounded, they may be slipping inside the gears. The tranny gear may be worn out. Or the speedo gear may be worn out.
Once you've assessed the parts, you'll know what needs to be done next.
The gas gauge is driven by a very simple potentiometer (resistor). A float attaches to an arm that has a wiper that travels along the potentiometer. When the resistance is highest, the gas tank is lowest and vice versa.
You can simulate this behavior by touching the wire connecting to the fuel tank sending unit to ground and then releasing it. If the gas gauge doesn't move, there may be a problem with the wiring or the gauge. If the gauge moves, the problem is probably the sending unit or its ground connection.
The key to troubleshooting problems with systems is to learn what all the parts of the system are and figure out a way to isolate each part of the system and confirm that it's working or not working.
