Disclaimer: This is based mostly on my own experience.
着く{つく} is the most commonly used verb for "arrive", as in "I've arrived at the station".
届く{とどく} tends to be used for things or abstract concepts. For example, on instant messenger, if you send a message and you want to check if the other person got the message because you didn't get a response, you would use 届く. You can also use it for feelings; it is commonly used for love, e.g. 私の気持ちが届いたでしょう, which can be properly translated as "I wonder if my feelings arrived?", but more loosely as "I wonder if you understood my feelings (i.e. I love you, etc)".
到着する{とうちゃくする} tends to be used for long distances and/or for long-distance modes of transportation, and especially most commonly for airplanes. For example, when your airplane lands, you may send a text message to your friends/family saying 到着した.