I have neglected many rights of Allah in the past, such as prayers, fasts, and zakah (paying poor-due). Am I required to repeat them or to compensate for them?
As far as the five daily prayers are concerned, the preferred opinion is that the repetition or compensation (or what is known as qadha in Arabic), is not obligatory for the reason that prayers are obligatory at a particular time, and once that time has elapsed, there is no way to offer them. However, one should seek Allah's forgiveness and offer as many nawafil (voluntary prayers) as possible.
As for fasting-yes; if a person was a Muslim at the time they became obligatory and he did not fast, then their performance is required, accompanied by the feeding of a poor Muslim for every day of fastng that was not observed during previous years. What is required is always a day for a day-no more, no less.
Example 1: A person did not fast three days of Ramadan in the year 1400 A.H., and five days of Ramadan in the year 1401 A.H., all out of negligence. He must now fast eight days, and feed eight poor Muslims in compensation.
Example 2: A girl reached puberty in Ramadan of the year 1400 A.H. but out of shame and modesty did not inform anyone of her menstruation, and, rather, continued fasting say for eight days of her period. Thereafter, she repented. She would then be required to redo eight days of fasting.
It may also be pointed out here that this is the predominant position. A minority opinion is that the ruling for fasting is the same as that for the prayers, i.e., there is no qadha (making up) for fasting in the case of a person who did not fast and did not have a valid reason for not fasting.
As for he who did not offer the obligatory zakah, it is necessary for the repentant to pay it for all previous unpaid years, for it is, from one angle, Allah's right, while from another, the poor people's right. (See Madarij Al-Salikin for further details).
And Allah knows best.