If I want to start buying domains, what do most people do? If I want to do private, what is the best and or cheapest site to go with?
If your offline business already have a strong brand name in the market then go with the real business name. Domain name with keywords no longer give much weight as before after google algorithm punish on exact match domain that do not have quality content.
Most of the time, it doesn't really matter if you're starting the business from scratch. But as was said above, if you have a strong brand name already, you're better off putting it anywhere you can to build up traffic to your site.