I am proposing a system using existing pumps and lines to transport water to an area that now requires this water in our factory.
For simplicity sake, this proposal utilises two centrif pumps in series, 20HP and 10HP although the first pump (20HP) would be sufficient for the task.
The system will be run manually. If the second pump is not running when the system is started, is there a possibility the second pump may start spinning in reverse as the water flows through it from the first pump? What then if the operator starts the second pump? Is there anything else to be wary about with a system like this?

Bypass the 10Hp pump with valves. If the 20 HP is running and spinning the 10 Hp pump backwards all manner of bad things can happen, the impeller comes loose, the 10 Hp motor starts and tears up the 10 Hp pump.  
If the 20 Hp can do the job isolated the 10 Hp pump until needed. You overcome all types problems, hydraulic, mechanical, doing this.
Much as I hate to say this you might just use a check valve in the discharge of the 10 Hp. But valves will be better as it allows you to isolated a pump for maintenance.

1. If the pumps are connected in series and are similar you probably do not risk a reverse rotation.

2. If the pumps are connected in parallel it is usuall to install check valves to avoid reverse rotation

3. If you want to operate both: check that the pumps flow are compatibles and that the suction pressure of the second pump is in the operational range.

4. If you operate the 20 hp pump and the 10 hp pump is off and (no by-pass) you may:
a)have higher presure drop on the line, the pump will be like an obstruction.
b) there is the risk of induction in the motor by the residual magnetism. this could damage the motor windings.

you could add a check as by pass for the second pump and this may work without any electronics or opening valves.

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