If you like terries you could try bamboo ones they are more absorbent than cotton
With regards to fast drying and absorbent, the more absorbent something is then it absorobs more water so therefore it takes longer to dry so you kinda have to compromise a bit there i am afraid. However when you look at a nappy if it folds out flat like terries then it is faster to dry.
Little lambs or tots bots are a good value fitted and fab for new babies they are pretty much the same but different fits so maybe get a couple 2nd hand to see which you like.
Of course there are more fancy fitted nappies more absorbent ones and prettier ones but then they become more expensive. still cheaper than sposies though
Then you have pockets and all in ones to consider. Have fun!