Hi Everyone, I am very grateful for your responses. It is really nice to know there is an active community for microcontrollers, especially since I spent some time this weekend figuring out what a transistor actually does..!
So, first I must admit that I used a 2N3904 instead of a 2N222. From what I gather I can go up to 1A on the 2N222 but only 200mA on the 2N3904... so that may be my first mistake. However, the pump is 3v-5v and 100mA to 200mA, so even then I should be good, no?
Pump: https://www.amazon.de/-/en/gp/product/B07QGW5HHB/ref=ppx_yo_dt_b_asin_title_o01_s00?ie=UTF8&psc=1
@wildbill - My understanding is that with these pump specs, the +5V output should be able to handle this as it is hooked up to a transistor. I know the IO pins are max 20mA, but the +5V pin is 500 mA when powered via USB and 1A when powered via VIN with >1A 5V Power Supply.
JCA34F - When the flat side is facing me, I understand it is EBC. But your diagram says the opposite. So I am not sure.
@ballscrewbob - I am going to run to the store and see if they have a MOSFET because it is something I didn't get in my "starter pack" and I read a lot about them this weekend. I will also pick up a relay. Good to have the right supplies to keep moving along the learning curve.
@groundfungus . Check on Flyback diode. Will hook up as pictured here, although I don't really get how that could work. I'll just take your word for it (for now). Luckily I have a few 1N4007s in my little kit.

I'm trying to figure out how you got to 180-220 Ohm resistor for the base. I'm seeing (on my PN2222 data sheet) that I have a Hfe of
minimum 35 @ Vce=10V, Ic = 0.1mA, and
minimum 100 and maximum 300 @ Vce=10V Ic=150mA
My (likely false) logic is that I need a maximum 200mA running through my transistor (Ic), and my gain is 100, then I need 2mA on my base current (Ib)? So 5V/.002 = 2500.
I'm looking forward to another late night! Thanks Everyone