Yes, you can go to any hosting company and search a desired domain to see if it’s available. However, those hosting company/domain name sellers (registrars) have a vested interested in grabbing domains that have been searched and selling them for extravagant prices.
There are many anecdotal instances where someone looked up their name with a company such as Godaddy and saw it was available. They go back a couple of days later to buy it, but it’s no longer available and now Google owns it! It’s still for sale. It’s just that now the price has gone WAY up.
So here’s what I do. I go to a website called WhoIs.net to check on my desired domain names. They don’t sell domains and I’ve seen no evidence that they buy up searched domains. They have no vested interest and are considered the standard domain availability search engine. So do your searches there!
When you do go to search your desired domain, be sure that you first make a list of alternative domain names that make sense for your business. This list is just in case your first choice is not available.
Try to keep the name as concise as possible. Make sure it’s easy for someone to say without tripping over the words. And make sure your words together don’t actually make an unintended other word or meaning. Could be very embarrassing.
When coming up with domain names (addresses), avoid dashes and numbers as much as possible. They are allowed for domains, but people just don’t think to use them when typing in a web address. But if there is a numerical digit in the name of your business, then using that number may be just the right thing to do. You want it to reflect your business and also be easy to memorize or type.
Okay. Happy hunting!