So the fabric dyeing went pretty well. The Kool Aid ones shown below came out just as I wanted them (except that I probably should have ironed them before I took a photo!), I used only about a quarter of a packet per piece of fabric so that the colours would be muted.
The tea dyed fabric didn't work quite as well, the colour isn't as dark as I'd like. I simply added about 10 black tea bags to a small tub of hot water, let them steep for about 10 minutes, took the bags out and added the fabric. I left them in there for about 4 hours but I think a better way to do it would be to use a large pot and keep the tea water hot throughout the dyeing process, or let them cool and bring back to a boil. I don't know why but I'm convinced you'd get better results with this method in less time.