We have the custom to get up at the break of dawn to recite Selichot and tachanunim from Rosh Chodesh Elul until Yom Kippur. (Rema: The minhag of Ashkenazi Jews is not like that. Instead from Rosh Chodesh on we blow the Shofar after Shacharith, and there are some places where they also blow the Shofar after Arvit. We get up at dawn to recite Selichot from the Sunday before Rosh HaShanah on; and if Rosh HaShanah falls on a Monday or Tuesday, we begin reciting Selichot from Sunday the week before Rosh HaShanah. It is forbidden for a mourner to leave his house in order to hear the Selichot at the synagogue, except on the eve of Rosh HaShanah when we extend the Selichot...)
We have the custom to fast on the eve of Rosh Hashana (Rema: And those that are exacting (Hasidim) are accustomed to fast every one of the ten days. And such is correct to do. And all these fasts are not necessary to do, and they are not announced on Monday and Thursday, even the (fast of) the eve of Rosh Hashana. If there is a circumcision on the eve of Rosh Hashana, one may eat. And most people are accustomed to eat on the eve of Rosh Hashana before daybreak; because the non-Jews are accustomed to fast on the eve of their holidays.
There is no tachnun on the eve of Rosh Hashana, even though there was in slichot. (and the shofar is not blown on the eve of Rosh Hashana).
One bathes and cuts their hair on the eve of Rosh Hashana. (and there are those that are accustomed to immerse in a mikvah on the eve of Rosh Hashana on account of seminal emission. And there are places where there is a custom to go to the graves and to pray very much there and to give charity to poor people there.)