I've been thinking about arranging an appointment with a therapist. What happens in therapy?

It depends partly on the type of therapy the therapist uses. But they all pretty much come under the broad umbrella called 'talk therapy,' if your problems are fairly general, like depression or anxiety or getting along with people. Some, like couple counseling, are structured for a quick set of sessions to learn techniques for communication.
Talk therapy usually starts with a sort of introduction of how the therapist operates and then the question 'How can I help you?' In other words, you define what you need and want to talk about, and the therapist guides the discussion with insights and suggestions, often practical ones. (Longer term therapies are sort of a thing of the past, in which the client talked and the therapist said very little, so as to not direct the process.)
Sometimes a referral to a psychiatrist is made, for medications (if you want). Sometimes a referral to a group is made after a while.
It's all up to you. You can quit any time you want, although you will be asked to phase out over a few sessions.
You have to agree to a 'contract' that if you feel suicidal, that you will call an ambulance, talk line, or the therapist. It's important to distinguish between thinking about it in general and planning it specifically. The therapist has a legal obligation to call an ambulance if you say you really are going to kill yourself. (Plus - they are human. It's tough on them when they lose clients to suicide.)
If you want, you can ask to interview a few therapists to see who you prefer. Nothing wrong with good vibes from the start, although that's not always going to mean that's the best person for what you need.
Many people develop a strong attachment to their therapist, regardless of gender, and that is called transference. It's considered good for working out difficulties with other people, especially past ones - parents, siblings, exes, someone who harmed you. You project your feelings about those people onto your therapist and go through whatever pain occurred and hopefully come out the other side. These are sort of old fashioned notions that modern therapy doesn't have time for, unless you can afford to keep paying when your insurance cuts out.
In general these days, it's all about working together on practical solutions for a set of problems that you both identify more clearly than you had before.