We have 3 environments: dev, test, and staging.
I want to check in and out of TFS. When we make changes, I want to promote the code to the dev web server. Next I want to promote the changes to test, then to staging. Would it be possible to do this with Team Foundation Server?
Why on earth are people suggesting branching? You branch for different features or static branches for release snapshots.
Surely the differences between these environments are configuration items/files and settings within these. All you need to do is get your deployment and release management process in order.
Create appropriate MSBuild tasks and use TFSBuild (Continuous Integration) to call these to take care of outputting the correct config files for the Environment/Configuration you are building. You can trivially add another MSBuild target that deals with the appropriate deployment to the respective target environments.