An excellent article on the things to do in the last ten days of Ramadan, and how to effectively spend Lailat-ul-Qadr (one of the odd nights of Ramadan):
Maximize the last 10 days of Ramadan
I'm posting a summary to this article, full text can be found here.
Here are some tips of things we can do on the Night of Power and the time before and after it.
One of the best times to do this is during the last part of the night. That means for instance, waking up one hour before Suhoor time to ask Allah for anything and everything you want that is Halal. This can be done using the Duas of the Sunnah, but also Dua in your own language, with sincerity and conviction.
10. Memorize a different Dua every night
They don't have to be long. They can be just one line. And be sure to know what they mean generally at least, even if you don't know the exact translation in English.
11. Attend the Dua after the completion of Quran recitation
Almost all Masjids where the Imam aims to finish an entire reading of the Quran in Tarawih prayers in Ramadan will be completing their recitation in these last ten nights. They may try to end on one of the odd nights and read the Dua at the end of a reading of the Quran. Attend this particular night's Tarawih prayer with your family. See if you can attend different Masjids' Tarawih prayers the night they finish reading the Quran.
12. Plan for the next year
Once you've done a self-evaluation, you can plan on where you want to go, at least in the next 12 months. Laylatul Qadr is a great night to be thinking about this (without taking away from your worship), since you'll Insha Allah, be in a more contemplative state. You may choose to dedicate one night of power for evaluation and one night for planning for the next year.
16. To do list for the Night of Power
Make a to do checklist for each Night of Power. This should define how you would like your night, the one better than a thousand months, to be used. Pick things from this list and define the sequence you would like to do things in. This will help you avoid wasting your time in unproductive chats which common in the festive atmosphere of Masjids at the Night of Power.