I started the ladder diagram, it's not all that hard to figure. helps quite a bit actually.
That is a good start.
Now how did you do the ladder diagram? Draw it by hand, or use some PLC software that you already have, or what? That will help us to set where you are in the PLC selection process.
Yes, this system is for my back yard.
That is a pretty good indicator that you need to power it from your 120 volts AC utility power. But it would be safer for home use if you could limit your PLC Inputs and Outputs to 24 volts AC. Also, you will find much more PLC equipment rated for 24 volts DC, so is it possible to switch your solenoids to DC versions?
If you can run everthing with 24 VDC, then you can buy a 120 VAC to 24 VDC power supply, and then buy a PLC that runs from 24 VDC and also has 24 VDC Inputs and Outputs. If the Outputs must be 24 VAC, then you will probably need PLC relay outputs (for a 24 VDC PLC) and a 120-to-24 VAC power supply.
EDIT: Reading back, I think you probably already have the 120 VAC pumps and the 24 VAC solenoids. So you need a PLC that can handle both of those voltage levels. The best fit is probably going to be one that has relay outputs. For the two pumps and the 2.4 Amp solenoids, you may need large external 10-Amp relays to handle the in-rush currents. Your PLC internal relay outputs would control the external relays (which in turn switch the larger motor and solenoid currents). Generally a PLC relay will handle about 1 amp, although there are some that go up to 7 Amps, but your pump starting current may easily exceed 3 Amps even if the running current is much less.
Look at the Automation Direct CLICK PLC. It is cheap, comes in 120 VAC or 24 VDC models, the three "standard" Click models have internal real-time clocks that can be used for control, and there are many Input and Output modules available. One of the output modules is the CO-04 TRS Isolated Relay module ($44) that has four 7-Amp relays. That might be just what you need to avoid using external interposing relays. You can use up to 8 Input and Output modules on the CLICK (in addition to the built-in I/O points).
http://www.automationdirect.com/adc...kable_Micro_Brick)/AC_I-z-O_-a-_Relay_Outputs
Probably the model CO-01AR-D is going to be your best CLICK PLC fit. It has 8 AC Inputs and six 1-Amp relay outputs ($109). Buy a CO-01AR-D PLC, 3 or 4 of the CO-04 TRS output modules, a 24 VDC power supply, the programming cable, download the free software and free manuals, then you are in business!