Have I got enough pins in my Uno or should I use another board...?

Hi Ron,

JonRon:
a display, presumably this is a parallel connection?

Probably, but post a link to the kit or a picture of the display.

JonRon:
I'm using DHT22 digital sensors and I'll do as you say with transistors to switch the pumps.

I'm surprised Bob (CrossRoads) didn't suggest this (he usually does) but a TPIC6x595 (where x is A, B, C or D, depending) might drive your 8 pumps with only 3 pins. But you will need to build a shield or a separate stripboard or something. Are you up for that or looking to avoid soldering?