Which route should I take?
A) Using a SharePoint 2010 Visual Studio Project
B) Building a normal asp.net website and incluing my pages using the "Web Part Page" web part in SharePoint
Basically I'll be doing some workflows, like filling out travel expenses forms, with approval and feedback.
If you have MOSS 2010 Enterprise, you can utilize InfoPath forms, which are specifically designed for use cases like yours. See http://blogs.msdn.com/infopath/
HTH
Filling out forms, approval workflow, comment fields, it is in some form or other all possible out of the box; so I'd say before you use either A or B, look into option C: what SharePoint allows you to do OOB.
If you can't make what you want with OOB features, I would always go for option A.
As already pointed out, you can use InfoPath forms if the production environment is having Enterprise edition. In case you have only wss, you will need to develop workflows with aspx pages.
Ideally, workflows are recommended for such tasks. Though you can also accomplish the same using a custom webpart, its much easier doing the same with Workflows.