Can I go to a dentist to get my teeth all cleaned, all plaque removed?

Yes.. good start. Most dental offices like to first see you for an exam and x-rays to evaluate your oral health. They will than diagnose the health of your teeth and supporting tissues. If your gum and bone are healthy, a regular cleaning can be performed by your dental office. If you have signs of severe periodontal disease your dentist may have to refer you to a specialist (periodontist).
Yes. This is a routine service for all general dentists (or their dental hygienists) and periodontal offices. They use hand scalers, ultrasonic scalers and polishing cups or brushes.