I am not an attorney and this is not legal or "professional" advice.
Copyright protection will be the easiest, as an artist's work.
However, trademark is what you will really want. And that is not at all as cheap.
Overall, my advice is for you to get moving, quit worrying about what theoretical other people will do, and work on yourself. Otherwise, you'll remain stuck in "analysis paralysis".
If the logo represents a trade or service mark, the cheapest ways to ensure you're protected if some one uses and attempts to claim ownership of the logo are: make a letterhead using the logo and place a letter on the letterhead declaring yourself the owner and creator of the logo and the date the letter was written, place the letter in an envelope addressed to you, then mail it back to yourself via registered mail and sign for the envelope when you receive it. Store the letter unopened in a safe place. The letter's date and the postmarks on the envelope will be proof of date of creation and first use, which can then be presented in a court of law in case of dispute. The next step: register your logo as a trademark or service mark at the state level. Third, if you intend to do business in more than one state, then register the logo as a trademark at the Library of Congress. All of these steps affirm what is legally called "rights of first use" and "statutory trademark rights". Remember, you can't copyright or trademark an idea, but you can copyright or trademark a written work or graphic image expressing that idea.
Unless you have a mid size and up company, this probably isn't the best use of your resources. Are you running a startup? It's rare that a company will blatantly steal another company's logo.
If you must get it protected, services like LegalZoom can do it pretty cost effectively.
Happy to jump on a call and talk this through with you more if you'd like.