Both verbs have the meaning of motion.
Let's look at the following sentences:
(1) Я прошла парк мимо (Or: Я прошла мимо парка)
(2) Я прошла через парк
(3) Я перешла улицу
(4) Я перешла в другую аудиторию
"проходить" has many meanings, but here I will focus only on the two that were illustrated by sentences (1) and (2).
First meaning is "to pass by". In sentences (1), you do not enter the park, you are only passing it by. Foe example:
Я прошла мимо парка, я не знаю, убрали ли там мусор.
Second meaning is "to go through". In sentence (2), you enter the park, go through it and then leave it. You want to use this verb if you want to focus somebody's attention on the fact they you have actually been in the park. For example:
Я прошла через парк, и там было чисто, наверное, весь мусор убрали утром.
"переходить" also has also a whole lot of meanings, and again I only want to focus on two of them (sentences (3) and (4))
The first meaning is "to cross". In sentence (3), you may or may not actually step into the street, what is important here is that you were on one side and ended up on the other side. You might have crossed it by a bridge just as well. It is used with objects that look like lines (a road, a river, a railway, a corridor). You cannot really use word "переходить" with "парк" unless the park is known to be long and narrow, and even then it is questionable.
The second meaning is "to change location by walking". In sentence (4), it means that you have walked from one lecture hall into another one.
For the rest of the meanings of these two verbs, you should probably read the definitions here and here and then ask your teacher further questions. Russian verbs of motion is a huge topic.