Groove, Zoomcat; based on your words/drawings - while those ideas will allow direction control of a motor (and in Zoomcat's case, limit switches to shut it off) - neither of these will work for the OP's application.
What's being built is an ROV (underwater remote operated vehicle), the up/down (and left/right) motors need a way to be turned off (and while Zoomcat's limit switches could work in a steering control or similar, they aren't much use for a prop thruster).
What is needed (if using relays) is an h-bridge constructed from 2 SPDT relays (each with their own transistor control), one for each motor. Alternatively (and better from an amperage point of view) would be to use 4 SPST relays, one for each leg of the h-bridge; and 4 transistors (and diodes, and everything else) - plus you would need to be careful not to set the wrong relays creating a direct-short...
Really, it would be better to (and cheaper), depending on the current requirements of the motors, of course - to buy some IC-based motor drivers (L298, etc) instead; depending on the source, it would only cost a few dollars per motor, whereas the relay method could cost much more than that.
Of course, if the current requirements are high, then using relays may be the cheapest method per motor; high-amperage pre-built motor drivers (greater than 5 to 10 amps) get expensive quickly, whereas relays are cheap.
The big disadvantage, though, of relay h-bridges (unless you have a beefy mosfet controlling the motor power input), is the inability of them to PWM speed control the motors - this applies whether the h-bridge is driving a low current or high current motor (though it can be done with a low current motor if a suitable transistor or mosfet of the proper size is used to control the motor power input, as mentioned; a large NPN or N-channel mosfet on the low side of the h-bridge and ground could work).
The OP's gonna need a bigger allowance to complete this project if he wants to use the Arduino in it!
;D