Can Arduino control 8 solenoids?
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.