The best approach really depends on what exactly you're trying to achieve. So far, this is what I can understand from your question and the way it's written:
- You have no experience with Tridion, given how you mix TOM.NET, CoreService and Content Delivery in the same question. TOM.NET and CoreService are Content Management APIs, not Content Delivery
- You say you want to access the data in the Tridion Broker Database using .NET. If the data you're trying to read is contained within a page published to the file system, access to the database may not be sufficient.
- If all you need is metadata about the published content I would seriously recommend configuring an OData endpoint linked to the same database, and use this to talk to Tridion. The instructions in the link above are for a Java-based OData server, but you can also do the same with a .NET-based OData server, the Tridion installation files include a sample .NET webapp for this.
If you explain a bit more about your use case, we may be able to help you further. As is, this question is really not clear enough.
If all you want is API-level access, then follow Bart's recommendation.
Since you added some information, here's some more info from my side too:
The content you want to read is in the page content. Do you know if these pages are published to the database or to the file system? If they are published to the database, my recommendation to use the web service still stands - you need a license to run the webservice server, but not for the clients. If you run the API then you need a license for the machine running the code.