In its simplest answer you get credits by earning them.
You can do this in several ways:
1. The most common is to join a faction to get a regular paycheck
2. Use CP's to buy yourself a ship and sell them on the markets.
3. (Less likely) Hope for a benefactor to help you get started.
4. Use Cp's to buy yourself a ship, and offer services to other players.
Yes. mail them and ask for an assignment. Most factions start payroll after 10 days of activity while in the faction. Then at the end of the month you will get a pay check(credits will be sent to you).
Looking for donations for Give freelancers the Galaxy fund.