Depuis que

“Depuis que” is used to indicate that something that started in the past continues in the present. It is also used to state the time the action started.
It is followed by a subject + verb.
Examples :
J’habite ici depuis que je suis né. => I have been living here since I was born. (That’s when it started and it is still true)

Dès que

“Dès que” is used to indicate that an action ends right before another start. It has a meaning of direct following of actions. It can also be used for simultaneous actions, in the same way as “Aussitôt que” or “Sitôt que”
Example :
Je partirai dès qu’ils arriveront. => I will leave as soon as they arrive. (Once the first action is ended, the next one starts)
Dès que j’appuie sur le frein, la voiture ralentit. => As soon as I hit the brakes, the car slows down. (The first and second action are simultaneous in this case)

“Dès” or “Depuis” alone can be followed by a date, a noun for an action or a time.

À partir de

“À partir de” and “Dès” have the same meaning but “à partir de” is much more neutral. There is a much lesser meaning of direct continuity between actions.
Example :
J’arrête de fumer à partir de demain. => I will stop smoking tomorrow (I am not quite sure about my english translation here).