If it is critically important to preserve the behavior your application or if it is mission critical then conversion is the only path.
You need to redesign your monolithic applications so it has a least two layers. The User Interface, and a ActiveX DLL with your business logic. Better yet you have three layers, the UI, the Business Logic DLL, and a your Personal Framework DLL. For database applications the personal framework is often the Database and Reporting API the software is using.
Once you have your application redesigned. You can then rip off the UI and replace it with .NET. Once you have that tested you can then convert the business logic, follow by your personal framework. Luckily both VB6 and and .NET can use COM Components. If you put your business logic and your personal framework in a ActiveX DLL then you can reuse it in .NET.
The reason for this is so BEHAVIOR is PRESERVED. Your users expect the software to work in a certain way. Each intermediate step leaves you with a working piece of software working the same way it did in the past. If isn't then you didn't add enough tests.
It helps to implement unit tests. http://www.nunit.org. This way you make sure that the both VB6 and .NET version produce the same results and behaviors.
I recommend going from the top down as most of value of your application lies in your business logic and your personal framework. Note that a fair amount of people have their UI and business logic considerably mixed. In my company application the UI is a thin shell that calls a ActiveX DLL (not Control) that performs the actual operation. The more thin your UI layer is the easier the conversion.
This involves a lot of extra work over what most people think a conversion process involves. I have done over six major conversions of our company's software. We write CAD/CAM software that controls metal cutting machines. This software has been continually developed since 1983. Originally done in HP's Rocky Mountain Basic on a 300 series workstation is now under the .NET framework running under Windows Vista.
Each time I learned something new about the conversion process. The traditional idea of sitting down and doing a rewrite leads to a situation where it is 90% done after a year and then you chase that last 10% for the next three years.
By doing it methodically by starting with a good design, unit testing, and so on you will have a project that will 100% done in two years instead of a project that mostly done in a year. Note that the time estimates are relative. You have the bonus that each step leaves you with working software. Plus if your approach for the current step is way off base you didn't waste all the work on the previous steps as compared to doing it in one go.
I recommend reading up on refactoring which involves modifying the design of your code in a predictable way. Examples is adding a parameter to a class method or renaming a function. Read up on Desgin Patterns. Design Patterns are to objects as Algorithms are to function and data structures. A good design pattern will tell you how to create a series of object to achieve some commonly done task.
Hope this helps.