Both, in most cases, and often in that order. The question is whether anything in your history flags a medical condition that needs ruling out.
Start with your GP if you are in the UK. They will refer you appropriately. Generally an ENT is the right first stop if any of the following are present: sudden hearing loss, one-sided or pulsatile tinnitus, tinnitus with significant dizziness or balance problems, ear pain or discharge, or a history of head injury. The ENT's job is to identify or exclude conditions like otosclerosis, vestibular schwannoma, Ménière's disease, middle-ear problems, and so on. They may organise imaging.
An audiologist is the right person for the audiogram itself, for hearing aid fitting if indicated, and for the long-term management of the tinnitus once medical causes have been excluded. Most of the actually useful day-to-day care — sound therapy, hearing aid optimisation, structured tinnitus counselling — sits with audiology, not ENT. In the UK the British Academy of Audiology has a find-an-audiologist directory.
Clinical psychology gets added if distress is severe.