How you find a publisher:
- Understand what the genre of your book is.
- Go to a book store or library and find other books in that genre that were published in recent years. Open them, look at the copyright pages, and identify the publishers.
- Find the websites of those publishers and look at their submission guidelines.
- Follow those submission guidelines exactly.
You can also find and submit to literary agents. In some countries publishers do not accept direct submissions. You find good literary agencies either in the books by the authors that write in your genre (sometimes on the copyright page, sometimes in the acknowledgements) or by researching it online (e.g. search for "name of author literary agent OR agency" on Google). Then go to the website of those agencies and follow their submission guidelines.
Publishers and agents that work with authors whose books you find in a library or bookstore are trustworthy.
Yes, traditional publishers take care of everything (editing, design, publishing, marketing etc.). You may have to rewrite (parts of) your novel, though, for them to want to publish it, so be prepared to discuss this.
Often the submission guidelines will tell you that the turnaround time is for them. If not, you'll likely have to wait a few months for an answer.
Under no circumstances should you choose an agent or publisher that you have to pay!