Here are the two ways I'd suggest that you create a SQL Server database:
1. If you have installed VS then you likely have installed LocalDB as well, which is a stripped down version of SQL Server Express. You can confirm that it is installed and install it if it isn't by running the VS Installer. To create a database, right-click your project in the Solution Explorer and select 'Add' -> 'New Item'. In the 'Add New Item' dialogue, select 'Common Items' -> 'Data' on the left and then 'Service-based Database' on the right. After naming you database and clicking the 'Add' button, and MDF data file will be created and added to your project. That's the database created. You can then double-click that MDF file in the Solution Explorer to open a connection to the database in the Server Explorer. You then have a similar (although not the same) interface as SSMS provides. You can right-click the 'Tables' node and select 'Add New Table' and go from there to create the schema.
2. If you install SQL Server Express separately to VS then you can install SSMS as well. You can then open SSMS and connect to your SQLEXPRESS instance, then right-click the 'Databases' node in the Object Explorer and select 'New Database'. Once the database is created, you can build the schema in the same way as before.
Note that, in the first case, the MDF data file is part of your project and gets attached to a SQL Server instance at run time. That means that the database is deployed with your application and attached to the user's local SQL Server Express instance when they run your app. In the second case, the database is permanently attached to a SQL Server instance and is not part of your project or application. That means that it needs to be created independently of your application when you deploy. There are a number of ways that that might be done, e.g. SQL script(s) or restoring a backup, and you can do it on a single server and have every user connect to that one database.
Your connection string will differ in each case as a result. In the first case, the 'Data Source' will be something like "(LocalDB)\MSSQLLocalDB" and the 'AttachDbFilename' will be the path of your MDF data file. In the second case, the 'Data Source' will be something like ".\SQLEXPRESS" and the 'Initial Catalog' will be the name of the database. In both cases, you may well have to change the connection string once the app is deployed.