Newbie transistor question, trying to understand
I think I know your issue. If I understand correctly, this is your current circuit:







simulate this circuit &ndash; Schematic created using CircuitLab

The problem is that there is a PN junction between the base and the emitter of an NPN transistor. This means your maximum \$V_{BE}\$ is approximately 0.7V. So, if you measure 1V at the base, there must be approximately 0.3V at the emitter. The way to fix this is to move the transistor so it is low side switching like so:







simulate this circuit

It is important to keep in mind that BJTs are current-controlled current-sources. That is to say, a collector current flows based on a multiple of the base current, \$i_B\$. This multiple is called the current gain, and it's symbol is usually \$\beta\$ (Greek letter Beta). In this case (assuming your arduino is outputting 3.1V), you will get a base current of \$\frac{3.1V-0.7V}{1000\Omega}=2.5mA\$. 

Since the typical current gain of a 2222 is about 150 (I'm going off memory here), you can rest assured that the collector current will be high enough, as the transistor will be fully in saturation (all the way on).