If you think about it logically Yom Kippur should come first. First we should do teshuva say viduy, fast, etc (all the things we do on YK) and then we would be prepared to be judged, namely Rosh Hashana.
The answer is as follows. For a person to do teshuva they need to understand that they did an aveira and that there is someone who cares, Hashem, and that Hashem can forgive them. Until a person comes to that realization it is futile to attempt to do teshuva. Until you realize that there is a King of the World who cares what happens and is involved in the world, you cannot do teshuva. Therefore Rosh Hashana comes first. The theme of RH is that Hashem is King of the world. We go on about Hashem rules the world and judges people. Once we accept that Hashem can judge us for our aveiros and punish us, then we are ready to do teshuva. We can then understand the severity of our Aveiros and realize that Hashem is their to forgive us if we do teshuva. This can only happen after we accept the din of Rosh Hashana.
I hope that we all accepted Hashem's kingship on RH and are in the process of doing teshuva so that we hava a gmar chasima tova.