Are priests a part of apostolic succession? I mean do they keep a record of every priest all the way back to the apostles?
Short answer: No. The Apostolic Succession is traced through the Bishops, who possess the fullness of the priesthood. From:newadvent.org/cathen/12406a.htm
This priesthood has two degrees: the first, total and complete, the second an incomplete participation of the first. The first belongs to the bishop. The bishop is truly a priest (sacerdos), and even a high-priest; he has chief control of the Divine worship (sacrorum antistes), is the president of liturgical meetings; he has the fullness of the priesthood, and administers all the sacraments. The second degree belongs to the priest (presbyter), who is also a sacerdos, but of the second rank (“secundi sacerdotes” Innocent I ad Eugub.); by his priestly ordination he receives the power to offer sacrifice (i.e. to celebrate the Eucharist), to forgive sins, to bless, to preach, to sanctify, and in a word to fulfil the non-reserved liturgical duties or priestly functions. In the exercise of these functions, however, he is subject to the authority of the bishop to whom he has promised canonical obedience…
Additionally, only a Bishop can make a Bishop or a Priest - a Priest can do neither. For this reason, among others, we trace Apostolic Succession through the Bishops.
If you have further questions, please ask.