Round 1 is your chance to introduce yourself to the Judges as an artist in one minute and thirty seconds. You will want to have a clear understanding of who you are as an artist and what you have to offer that is unique. You will want to sing a cut of the song you are most comfortable with, which also gives the Judges a sense of your range and personality, and which is as impressive as possible so that you stand out enough to make it to Round 2. I would suggest singing something with high energy for the first Round so you grab the Judges' attention right away.
Round 2 is your opportunity to one-up yourself: to sing something even more impressive than you sang in the first round, AND to show the Judges another side of your voice or personality you did not show them in the first round. If you sang something upbeat in the first round, this is your chance to sing a ballad or a song that shows both slow and upbeat singing, as well as soft and loud singing. If you didn't showcase your riffing in the first round, this would be your chance; if you didn't pull an impressive high note out of your hat last time, this would be the time to do it. This is your last chance to let the judges know all the tricks in your bag and what kind of a performance they can expect from you in the Finale. Make sure you present yourself consistently - in other words, don't sing a musical theater song in the first round and then whip out an operatic aria in the second round. That is confusing to the Judges. If your act includes a mashup of songs in multiple song genres, make sure you do that in the first round so you aren't confusing the judges in the second round. They are looking for certain types of acts to showcase, so you need to stay consistent. Also, there may be a new judge in this round who has never heard you before, so keep in mind that this round will be another first impression for someone.
The Finale: If you make it to the Finale, it means the judges are excited about what you have to offer in an exciting showcase of talent of all kinds. This is your chance to WOW not only the judges, but also the audience, who will be 50% of the vote. There will only be ONE judge who has seen you through the entire competition, and the rest will have never heard you before. This is your chance to whip out your best tricks, your best song, your highest energy, your most virtuosic performance. Sing what you sing best BUT also: CHOOSE A SONG THAT IS A TOP CHARTING HIT SONG (either now or in the past) THAT IS EITHER HIGH ENERGY OR BREATH-TAKINGLY BEAUTIFUL. That is what makes a good Finale song. The most successful songs are songs the majority of the audience LOVES, a song that is currently relevant and popular (like the hit songs that come out of recently released hit movies), a song that shows virtuosity of some kind (a powerful climax, an awesome high note, a dramatically intense moment, a stunning vocal quality), and a song that either takes the audience's breath away, makes a little kid say, "Wow!" or makes them want to get out of their chairs, dance, and give you a standing ovation! Make this the most amazing performance of your life!