If this is, indeed, a PN2222, and if your Fritzing is a true representation, then your transistor is connected wrong.
If you pop it out, and turn it 180°, and plug it back in, it should be facing the right way.
As for your "5V across" comment, when juxtapositioned with your resistor fiddling, it sounds like you have a misconception about how a Bipolar Transistor is "biased".
The thing to know is that a bipolar transistor is a "Current Amplifier". Thus, it's the current on the input, that makes it "go". If you try to put 5V across the Base-Emitter junction, you will either fry the transistor, or you will severely tax whatever is trying to supply that 5V -- in other words, somebody gonna suffer! In fact, if you try to do that with an Arduino output, there's a good chance you'll burn it out!
OK, so what is the proper way to bias a bipolar transistor? Depends on how you want to use it, but lets just talk about using it as a "switch". Like I said, before, a bipolar transistor amplifies current. So, to figure out what current to apply on the "input" [the Base-Emitter junction], we need to start with how much current we want to switch. In your case, that will be whatever maximum current that motor will be drawing -- and the maximum current will, invariably, be the start-up current [and actually the stall current, which will be nearly the same]. So, first of all, I'm going to assume you meant a "2N2222A" [slightly better than a 2N2222], which is rated a 625mW MAX, so rule of thumb, half that for, lets say 300mW.
Figure 4, of the ON Semi datasheet, shows around 210mV for a Collector current of 500mA, when the base current is at 40mA, the ABSOLUTE MAX current an Arduino output should source. That amounts to a power dissipation of:
P[sub]D[/sub] = 500mA * 210mV = [b]105mW[/b]
So, well within the 2N2222A's 300mW power handling capability. So, that would be the ABSOLUTE limit of using this transistor, with an Arduino, as a switch. I, personally, don't like demanding more than 20mA from my Arduino outputs 
So, back to biasing that transistor! Let's say your motor needs a half an amp to get it running. And, let's say we don't mind torturing our poor little Arduino, by forcing it to supply 40mA to the base of that transistor, then, the math would be as follows:
R[sub]Base[/sub] = (5V - VBE)/40mA = (5V - 0.7)/40mA = [b]107Ω[/b]
-- or next standard value up:
** **110Ω** **
Now, a note or two about something called Beta, and how that figures into the relationship between drive current [what flows through the Emitter-Collector junction] and Base current [what flows from Base to Emitter]. Beta [often, the Greek letter
** **ß** **
is used to indicate "Beta"] is the ratio between the Collector-Emitter current, and the Base-Emitter current. There is the Maximum Beta, called the hFE. Then, there is the Beta for setting optimal conditions for switching. Most datasheets recommend a Beta of 10 for switching [sometimes it's a little more, like 15 or 20]. What's going on, here, is, you want the Collector-Emitter "saturation" voltage to be as low as possible ['cuz the higher that voltage is, the more power gets lost in the transistor, and thus is not delivered to the thing being switched]. And, the higher the base current is [up to, of course, whatever current will fry the Base-Emitter junction], the lower that Collector-Emitter voltage is going to be. BUT, there is a tipping point, where a large amount of current, into the Base, isn't going to make a significant difference in the Collector-Emitter voltage, and it becomes a matter of silly waste.
So, that tipping point is, generally accepted to be a Beta of around 10. So, if you need to drive something that needs 2A, then you'll need around 200mA on that Base. At, 1A, the Base will need 100mA, etc. [dare I say it? It's all about dat Base -- ooooh! Couldn't resist!!]
And, that, my friend, is why, if you need to drive something that needs more than around a half an amp -- consider using a MOSFET.
BTW: you might have noticed that in my little thought experiment, I set the Beta to:
[b]ß[/b] = 500mA/40mA = [b]12.5[/b]
!
Just wanted to see if you were paying attention! And, to show that you don't, necessarily, have to stick to a Beta of 10, even if the datasheet recommends it -- be an engineer, and fudge a little, now and then 
BTW2: In case you're wondering what that resister is actually doing -- it's, essentially, converting Voltage into Current. The Arduino output is a fairly constant Voltage that supplies varying current, as needed -- it, isn't, really, designed to deliver a specific current. The Base, on a bipolar transistor, requires a specific current -- which is way a conversion is needed. The resistor takes the Arduino's output voltage, and sets it to a specific current. That's the simple explanation. There are a lot of details, like the fact that the Output voltage isn't exactly constant -- it varies a little as the current changes, and then there's that Base-Emitter voltage, and how it behaves in the real world, etc. But, that's a whole different, rather complex, conversation, and actually, for your purposes, inconsequential.