Can Arduino control 8 solenoids? I know very little about Arduino and even less about programming one. Before I purchase a board, and learn how to program it, I was wondering if my idea is feasible.

I want to program the Arduino to sequentially open one after another irrigation valve. There would be approximately eight 24 volt solenoid valves. Each one would need to be open for approximately 15 minutes, then open the next valve in sequence and finally close the previous valve. Similar to a multi-zone irrigation controller.

I see that I may need to use a transistor in the power circuit of the solenoid. But it is not clear to me if the Arduino can control eight solenoids and if so, does each solenoid require its own transistor?
An Arduino can control (at least) 8 devices. Instead of a transistor, it might be much easier to use an 8 way relay controller, something like this:

8 relay module

Just connect all 8 valves to the relays, you power the relay with a separate source for the valves, and connect the pins to the Arduino.

These ones have opto couplers, which give an extra security between the valves and the Arduino. 

8 relay module with opto couplers
(note I just put the first entry I found, you might find better/cheaper similar items, e.g. at AliExpress for $4).

As Andre said in a comment below, you can use an Arduino Mega if you need more outputs. Or if you want a challenge you can use use a multiplexer IC like 74HC595 (plenty of examples to be found with Arduino).

For controlling one relay with an Arduino enough examples can be found, for 8 it's similar (just use 8 GPIOs), or a multiplexer IC.