I am reposting this (with some minor corrections/additions) from last year as I believe that this is a very important point to think about as we approach Rosh Hashana.
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), be forgiven for our sins, and then we would be prepared to be judged, namely Rosh Hashana. Instead we do things in what seems to be a backwards way. First we are judged and only after we were already judged do we have Yom Kippur which is a day set aside for teshuva and kappara.
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. You need to accept the authority/kingship of Hashem so that you understand that you did something wrong and that you need to do teshuva. Therefore Rosh Hashana comes first. The theme of RH is that Hashem is King of the world. We go on about how 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. Once we have accepted his authority/kingship we can then understand the severity of our Aveiros and realize that Hashem is there to forgive us if we do teshuva. This can only happen after we accept the din of Rosh Hashana and therefore Yom Kippur can only come after the din of Rosh Hashana.
I hope that we all accept Hashem's kingship on RH and are in the process of doing teshuva so that we have a kesiva vchasima tova.