Please note that you DO NOT want to play with large pumps. if you try to turn them on and off too often, you can burn out the relays or the motors. very bad. VERY expensive.
also, if you want to use an Arduino to turn on a large motor, you will need to have some special motor starter to do that work. you should be able to use a simple relay to send a signal to the starter. a MOTOR STARTER is a special device.
As I see this project, you have at least 4 phases here.
#1) software : get a toy pump to fill a bucket and prove the logic works. I use a tiny pump and a trashcan sitting on the floor with one pump and a smaller trashcan on a desk that automatically drains into the first tank.
#2) high current, high power AC (?) electrical circuit design
#3) low current DC electrical circuit design
#4) interfaceing the DC to the AC and troubleshooing
#5) physical installation , hoses, valves, sensors, wiring, etc.
WE would ALL advise you STRONGLY to post a picture of your proposed installation. a simple sketch of levels and pump numbers and switch numbers and then to write a sequence that YOU would do if YOU were to do it all manually.
1 check if lower tank has enough water to pump with
2 check level of next tank
if needed, turn on the pump
monitor the tank level
turn off the pump once the level is reached.
YOU would do that.
and you can program that pretty easily.
if you can do that, we can have a better understanding and help more easily.
but be warned, as we know more, we will want more detals. motor type, voltage, current, how it turns on and off.....