Intake to a mental hospital is usually done in the Emergency Room. You can expect it to take several hours as you'll need to meet with each of the intake team members individually and you'll probably have to wait quite a long time for each one. Each of them will probably ask you the same questions. Be patient. Most likely they're not communicating with each other anyway. And they'd like to get the information straight from you rather than from someone else.
You will meet with a physician, a social worker, and several other people whose roles I don't recall at the moment.
Remember, you can pick which hospital you want to stay at. If there are no beds available at the hospital of your choice, enter another one with the understanding that you will transfer as soon as a bed becomes available at the hospital of your choice.
If you need to be transferred to another hospital, you will be taken via ambulance.
If you do need to transfer, be careful with your belongings. I have lost personal effects during a transfer.
Most importantly, remember that you have rights as a patient. Don't let anyone tell you otherwise.