Actual lawyer here. This information is generalized as pertains to the US, but can vary a bit from location to location:
For the vast majority of crimes, when you are arrested you will be taken to the city or county jail, processed, and placed in holding. Holding is generally a large cell, filled with other individuals awaiting their first day in court. There will usually be a phone in this cell, with either a phone book or a bunch of business cards for local attorneys and the public defender. If not, it will be near the holding area and the guards will give you access as needed. The "one phone call" is something of a myth, unless your guards are incredibly unprofessional. You'll be allowed to make multiple calls to call an attorney or call your family to have them get an attorney.
Eventually, you will be taken to court for your arraignment. Most states require arraignment within three business days, some two. At the arraignment, the judge will explain what charges are against you, tell you your rights, ask how you plea, and ask if you wish to hire an attorney or have one appointed. In most jurisdictions, there will be a representative from the public defender's office present and they will help you understand basic matters even before they are appointed.
If you elect to have an attorney appointed, you will be asked to fill out a simple form describing your income and expenses. If you're poor enough (lawyers are expensive so almost everyone is), the judge will appoint a public defender. However, the judge will also advise you that, if your income changes, you may be required to reimburse your jurisdiction for all or part of the cost of the attorney. The judge will then give you some papers with a summary of what happened at your arraignment and your next court date. If you want to find your own attorney, the judge will just give you those papers and tell you to have your attorney present at the next session.
If you have been hauled in for an interrogation, there will usually be signs posted explaining your right to an attorney and listing a phone number for the public defender. Simply tell your interrogators "I want an attorney." Not "should I have an attorney?", "maybe I need an attorney", "I think I should have an attorney", or even "can I have an attorney?" In many cases, when you leave ambiguity in there, the interrogators can claim your statement was rhetorical and that you didn't mean to invoke your right.
Keep in mind, the right to an attorney doesn't kick in unless you are arrested and being questioned or questioning is about to begin. So, if you voluntarily go to the station to answer questions, you don't get to demand an attorney. If you've been arrested, but no one is questioning you and you shout "I want an attorney", you don't get an attorney and later officers are free to question you unless you re-invoke the right.
Once you have invoked the right, no one may question you until you've spoken to an attorney UNLESS you re-initiate questioning.
That should answer your question and then some. Keep in mind this information is generalized and not meant to be advice on how you should handle your specific situation.