If you like the Bamboozles, I'd agree with previous poster the new Stretch have a microfibre core so this aids the drying time, and makes it similar to cotton drying speed. The size 2 have poppers down the front so you canr educe the heght as a one size - the popper in booster comes with it so you can add that in as required later (for now it keeps the bulk down)
A slightly quicker drying alternative would be a microfibre nappy such as the Nature Babies MicroDiddy - it's lined with cotton jersey so you have cotton next to the skin, and the second size is a one size so you can fold the front down and use as a birth to potty. It fastens with nippas so you can get a good snug fit.
I'd leave buying any extra boosters until you need them. Some babies do for overnight, some do all the time as a toddler and some are quite light wetters and really don't need them at all. Bamboo is slimmer for similar absorbency as cotton but slower to dry. You could even make your own from cutting up an old towel.
The Bummis wraps are good BUT they can be shallow between waist and thigh above the leg - so do be careful if you buy a nappy that is bulky at the waist (one size!) as you may have trouble getting it to fit under if your baby is chubby (less of an issue if skinny). Another very good wrap is the Nature Babies Classic wrap, they also make a budget Essentials one size wrap too - very reliable and very good value.
If you aim for about 20 nappies and 4 wraps per size this will keep you in enough nappies to allow for drying. A few extra flat nappies (prefolds, terries) and a nappi nippa never come amiss (cheap + quick to dry) to add extra to help you with drying
A wetbag is useful for out and about but equally you could just use an old supermarket carrier bag. I used to just pop the nappies in when dirty tighten the drawstring to keep it all contained, and take them home, then wash the bag and the nappies.