I dyed all my diapers and it was great fun! I used Dharma dyes and they work wonderfully. Not cheap but worth it in my opinion. Procion dyes like Dharma sells are the best quality dyes you can buy. I would recomend against anything like RIT, it's cheap but you won't get anywhere near the bright/vivid colors you will with procion dyes and it could run as well as fade much easier. Some also like Dylon dyes, I've never tried them but from what I gather they're somewhere between cheap stuff like RIT and the procion dyes.
Decent quality dyes shouldn't run if they've been rinsed enough, but this may take quite a few rinses depending on how much dye you used and the color of the dye (some colors are harder to rinse out that others - red is one of them if I remember correctly). You can tye-dye the diapers if you'd like, I did something similar by using the same dye recipe as tye-dye but instead of actually tying the diapers I just used a small dye-filled syringe to put the dye where I wanted it on the diaper. Be warned though that dye spreads easily in a diaper since it's so absorbant, so don't overlap colors unless you're going for a combined color. Also be careful which colors you put next to each other (for example blue and green don't mix well and just form a muddy brown/black). It's almost inevitable that the colors will bleed together during the process so use colors that will complement each other well, like yellow/orange, yellow/blue, orange/red, yellow/red, red/blue (will make a dark purple). And no, absorbancy won't be affected.
Unfortunetly I don't advise trying to dye covers. I have tried and had very limited success. The covers are polyester and it's very difficult to dye polyester, usually requires either extremely high heat or very strong chemicalsl; neither of which is doable for a home dyer. I dyed a couple of mine a very light pink using a fuschia dye, tried 'orchid' and got a very light purple and a light jade green (can't recall which dye I used for that). Tried red and got absolutly nothing. The colors I did have some success with are almost all available from ME now anyway so there isn't really any point. And even repeating exactly what I did may not work as I tried once with AIOs and only the inside diaper part took color, the outside remained white. This despite doing the exact same thing I'd done with my own covers when I got them purple and pink.
I have to say though, the ME colors look fantastic so unless you already have plain diapers and don't want to buy more, I'd just go for the ones ME makes (wish they were available when I dyed mine). Much less trouble and you could have a huge variety of colors without buying 10 different colors and doing 10 different batches.