You can buy fairly cheap motorized ball valves (that's the type of valve shown in the second picture you shared). That will simplify the project quite a bit. However, if you need precise control of the water flow rate the ball valve is not ideal because a small rotation makes a big difference in the valve aperture.
A needle valve will give you the best control, followed by a gate valve. The third picture you shared is probably a gate valve but it could be a needle. I haven't seen cheap motorized needle or gate valves but I also haven't actively looked for this sort of valve.
A valve with very precise control via a servo or encoder or stepper will likely be more expensive so if you need that sort of precision the DIY approach might make economic sense (as well as being fun).
To get started learning programming you can start working through the example code included with the Arduino IDE under the File > Examples menu. Use the associated tutorials:
and the Arduino Language Reference:
to help you understand what every line of code in each example does before moving on to the next example. You don't necessarily need to do all the examples, only the basic ones and then the ones that are relevant to your project. Most of the examples will only require some cheap extra components that will be useful to have anyway.