If you prayed without wudu by mistake, you have to do wudu and repeat your prayer. There is no sin on you if you were unaware of that. If you were aware that you had broken your wudu and that it was haram for you to pray in this situation, then you have committed a grave sin.

1. There are 14 pillars of prayer and there are 8 obligatory parts of prayer. There are many Sunnah acts of prayer, both words and actions. 2. The difference between a pillar and an obligatory part is that a pillar cannot be waived, whether one omits it deliberately or by mistake, rather it must be done. An obligatory part is waived if one forgets, and it can be compensated for by doing the prostration of forgetfulness (sujud as-sahw). Learn more in the detailed answer.

If you forget the first tashahhud, stand up for the third rak`ah and start the recitation of Al-Fatihah, then you should not return to the sitting position, but you have to offer Sujud As-Sahw before the salam.