A psychologist won't help you at all. Psychologists deal mostly with the research part of psychology, and they can also do neuropsychological testing, which isn't a treatment. I'm going to assume you mean something else. What will help you are a psychiatrist and therapist.
As for which one you see first, that's up to you. A therapist will, well, work therapy with you. A psychiatrist will help you with medications.
Here's what you can expect from either one: Questions. Both of them will have an initial evaluation which lasts between one and three sessions, typically. They'll ask you questions about your history, your childhood, your adolescence, and they'll basically gauge how psychologically healthy you are while also trying to come up with a diagnosis, if it's warranted. You'll get asked about your current level of functioning, your social life, and other things so he or she gets an idea of who you are and what you do. During the interview, expect to feel a chill run down your spine, because they are watching you the whole time to examine your behavior during the interview, for instance, to see if you're nervous, or agitated.
After that, if the psychiatrist says you could benefit from treatment, that's when you begin. Because you'll probably have told him you have a history of depression and suicidal thoughts, he'll probably start right away, but modestly. Psychiatrists like to start at the lowest dose possible for pretty much everything, however, they will make exceptions. For instance, if they want to put you on an antidepressant, well...you know those black box warnings they have on SSRIs and SNRIs? The ones that say that you may experience a spike of suicidal thinking when immediately starting? They want to avoid that, and for good reason, by putting you on a very low dosage to start out with. They expect you to take a more therapeutic-size dose later on. From personal experience, I can tell you that those warnings are not bullshit. This is because modern antidepressants have a special effect of increasing your energy before doing anything with your mood. See where the problem comes in? You'll have a spike of energy to act out on your suicidal thinking. This should not be minimized. It happened to me, and my dosage had to be stopped right away. You could get a frenzy of nervous energy, but you'll still be thinking of suicide. An impulsive suicide attempt is very easy at that point. This effect can last from a few days to a couple of weeks, at which point it will be safer to put you on larger doses of the medication.
Therapy's a different story, and I can't say much about it because it varies from therapist to therapist. They all have their different styles and specialties. Some have very little training, others have doctorates. It ranges from a Licensed Mental Health Counselor to a clinical therapist with a Psy. D. I would recommend interviewing a therapist yourself, hopefully you'll be able to do it over the phone, so you can ask them what their specialty is. Keep in mind that some therapists do not like to work with suicidal patients, and although they may accept you as a patient, they will be totally unequipped to deal with your suicidal feelings. I had one therapist that dodged whenever I brought up suicide; she would always change the topic. I always ask, during introductory sessions, "Are you comfortable working with suicidal patients?"
I think that's a pretty good introduction. Hope some of that helps.