I'm confused with the usage of "sign up", "apply for", "register". Which of the following sentences is acceptable?
Please give examples of each verb phrase.

1. I'm looking forward to the one-day trip to Keelung.
2. I'm willing to apply for the activity.
3. I'm willing to sign up the activity.
4. I'm willing to register the activity.
1. I'm looking forward to the one-day trip to Keelung. <-- You are implying that you have already registered for the trip. The sentence is correct, though.
2. I'm willing to apply for the activity. <-- Correct
3. I'm willing to sign up for the activity.
4. I'm willing to register for the activity.

Examples:

1. I would like to sign up for the competition.
2. I want to register for the test.
3. Can I register an account online?
4. He wants to apply for a driver's license.

I hope this helps!
Actually, I have not registered for the trip. I just received the message on my e-mail that my friend sent me. So, how to express the acceptable sentence in #1?