If you aim for the timer based solution then you can get a timer IC:
http://www.ti.com/product/LM555There are many companies that produce such ICs and they all end with the number 555. This IC can be used for a lot of things are quite cheap so you may wish to grab 10-20 of these for various experiments. This IC is an analog IC so there is no programming involved. You use resistors and capacitors to control the timing so you should get a collection of these as well.
Unfortunately the 555 is not able to tell the time precisely, but I don't think the plants will mind that the timing shifts a little. Also the 555 is not stable for long intervals so I suggest that you create a system that starts the water flow every hour and let the water run for a short time.
You can use another 555 to control how long time the water should flow for. You most likely want a potentiometer to be able to vary the duration.
You need a solenoid valve to control the flow. I would recommend something that runs on 12 V. This will mean that you can run the entire circuit on the same voltage as the 555 has a maximum of 18 V (at least the one I link to). You want a valve that is turned off when the power is off (so you don't flood anything if the power goes away). You need a diode across the solenoid in order to protect the circuit.
You can't control the solenoid directly from the 555 unless you can find a solenoid that uses very low current. Hence, you need something to drive the solenoid from the output of the 555. I would recommend a relay since these are simple to set up (you also need a diode across the relay in order to protect the rest of the circuit). For slightly more advanced solutions you can use a transistor (BJT) or a MOSFET (which is also a kind of transistor).
This should amount to a nice DIY project and fiddling with the 555s means that you'll be able to use the scope to monitor all its signals!