.NET Platform is not opensource, .NET Core is open source but .NET Framework is closed. In either case you are free to use the frameworks without any cost. The only licensing restriction is around building a competing framework for obvious reasons. So yes you can migrate from VB6 to VB.NET (any version but .NET 6 is recommended) without any licensing issues or costs.
But to build a .NET app you need a newer version of Visual Studio. That is where the cost comes in. VS has a free version called Community but it is designed for hobbyists and open source projects. You can use it to build apps for personal or open source projects. But if you're building a commercial app you are expected to buy VS Professional (or higher). You can see the full license here. Note that there is no difference between the editions in terms of what types of apps you build. The difference is in a few extra features in Pro and, of course, paying MS to use the software. Of course you could also do everything using just command line tools (which are free) but that is really painful.
As for cost it depends on who you get it from and what you want. You can buy VS Pro directly from MS but many companies buy VS Pro with an MSDN subscription. It costs more money but you get free upgrades to newer versions, WIndows OS and other software licenses for free (for development) and licensing becomes easier to manage. However it is a subscription so you have to renew every year. The renewal price is cheaper. You can see all the pricing options here.
In summary:
- .NET is free for use with commercial, internal and open source projects. At least one version of the framework is already installed on your Windows machine anyway.
- Migrating from VB6 to VB.NET isn't trivial and is going to require significant effort likely resulting in a rewrite. Plan accordingly.
- Visual Studio is the tool you'll want to use but it does cost money if you're building commercial software. For internal or open source software you can use VS Community for free.
- If you just want to test out VS then download and install Community. You can install Pro later. If you plan to buy Pro but want to get started now then download the Pro trial. You have some time to test it before you need the license.