Yes, your total price for trademark registration will depend on the number of classes. Most jurisdictions charge a base price for the application, which includes one class of goods and services. Each additional class then costs extra. This price varies depending on the country, from about €50 to €250 per class. If you decide to apply in one class first, you can't add more classes later; you can only file a new application. You can learn more about this pricing structure in this article.
Besides classes, the most significant factors influencing the total cost will include the country/countries of your choice and whether or not you are using a legal representative to file the application on your behalf.