To create a resource in Azure, go to http://portal.azure.com, set up an login to log in. Next you will have to open a subscription, which you may have to associate with a credit card. (I'm a little foggy on that point, since I only use free credits that I get with my MSDN subscription.)
The subscription itself does not cost anything, as far as I know, but about every resource you create will be billed against your subscription.
There are lots of services, but search for Azure SQL, and then select Azure SQL Database. There are a couple of options, and I think that for you will will not need anything more than Standard. Maybe you can even do Basic, which is very cheap. Don't worry too much that your initial choice may be the best one. You can easily scale your database up and down - that's one of the joys with the cloud.
If you expect the application to be idle most of the time, you could consider a Serverless option. With Serverless, you only pay for the database when it is in use. A drawback, though, is that you may have to wait when you want start using it, as it takes a minute or two to load the database.
Since you will keep a client on-prem, a word of caution: make sure that the application is not chatty. If 100 rows are to be updated, send all 100 at once, and not one a time. When you have the client on-prem and the database in the cloud, there is always a latency. You may have a very high bandwidth, but the speed of light will always set a limit.