Should I supplicate after the athan while I'm at home, or should I go directly to the prayer?

Praying at the beginning of the prayer time is superior. Hastening to perform the pre-prayer Sunnahs (optional prayers) and then the obligatory prayer achieves this virtue. Supplication (Du'a) can be made within the prayer, or one can supplicate, then perform the pre-prayer Sunnah, and then the obligatory prayer. Supplicating after the Adhan (call to prayer) does not negate the virtue of praying at the beginning of the prayer time, due to the Prophet's (peace and blessings be upon him) saying: "When you hear the Mu'adhin (caller to prayer), say as he says, then invoke blessings upon me, for whoever invokes blessings upon me once, Allah blesses him ten times. Then ask Allah to grant me Al-Wasilah, which is a station in Paradise that will only be for one servant of Allah, and I hope that I am he. So whoever asks for Al-Wasilah for me, intercession will be permissible for him." This applies to everyone, whether pure, in a state of minor ritual impurity, in a state of major ritual impurity (Janab), menstruating, old, or young, with the exception of one who is praying, in the lavatory, or engaged in sexual intercourse. There is no harm in supplicating after the pre-prayer Sunnah or before it, or within the prayer. The only exception to this is the 'Isha prayer, for which it is recommended to delay it.