I've been thinking this circuit over in my head the last few days, and I've had a look around but can't ind anything that confirms I'm doing it correctly. Could be my Google-fu failing, or, more likely, I'm not doing it right. I'm an absolute beginner at electronics, but I have been reading and watching Youtube vids, so I'm hoping I'm understanding this correctly.
Arduino connected to a 595 shift register, the outputs of which are connected to 2n2222(2?) npn transistors, which have LEDs "above" the transistors (high side switching?).
I've put together a little Fritzing layout circuit diagram thing, attached, and I'd really appreciate it if you could have a quick look (it's only small, honest) and tell me if I'm doing it right, and if not, what I should do?
The idea is to have more LEDs per transistor, and more transistors (and grouped LEDs) to other pins of the register, I've kept it at two transistors and two LEDs to make the diagram simple.
I appreciate any help any of you can offer!
You need a resistor in the base of each transistor to limit the base current and protect the shift register output pins.
Aha, thanks! (Though I was kinda afraid of that, because I've no idea what resistor to use between pin and transistor, would 500-1k ohm be about right?)
I'm chuffed I got the logic right! 
would 500-1k ohm be about right?
Yes.
I would put a 0.1uF ceramic capacitor across the shift register's supply pins.
I'm chuffed I got the logic right!
Not checked all the pins on the input side as the schematic was too small for me to see the pins.
The pin setup I learned from the adafruit shift register tutorial (which I've run successfully), I'm considering putting OE and MR to the Arduino too, just to give me more control. Being able to dim them down with PWM between changes appeals And, 500-1k-ish ohms it is!
Thanks for the advice, but I have another question, and it's an absolutel beginner one about the protection capacitor - I'm guessing it's positively aligned? (ie, cap +pin on the +5v terminal, -pin to G, rather than the "led fadeout" alignment?). Yes, that's a total face-palmingly basic question, but I'd rather ask than continue thinking wrongly 
Apologies for the double post, but I've amended the schematic to include your fixes. I've also saved i a bit bigger so you can see the pins on the 595. I know I've not occupied Q0, that was just for schematic simplicity.
Hopefully I've got it right, and I do appreciate both your help and your patience!
about the protection capacitor
No it is not protection it is decoupling.
I'm guessing it's positively aligned?
No you use a ceramic capacitor 0.1uF these are not polarised.
cap +pin on the +5v terminal, -pin to G(round )
Yes as close to the chip as you can get it.
Here is why you must use them in all circuits you make:-
http://www.thebox.myzen.co.uk/Tutorial/De-coupling.html
New diagram looks fine thanks.
Informative link, thanks, bookmarked it after I read it. Gonna have to get some ceramic capacitors, it seems...
Buy a bunch - they are cheap.
www.dipmicro.com or www.taydaelectronics.com
50V rated, if you get higher voltage rating the parts are much bigger.
