Ведь is a particle. There is a very useful book called Particles in Colloquial Russian which you might find interesting. It explains ведь, же, -ка and a lot of others in detail.

I was told that you should use тоже in reference to the subject and также when it refers to any other part of the sentence.

Я играю в футбол. Пётр тоже играет в футбол.
Я играю в футбол. Я также играю на гитаре.

The rule seems to apply to the logical subject, however, not necessarily the grammatical one: мне тоже, у меня тоже... Apparently тоже tends to refer to the element preceding it, and также to the element following it.