How Do I Run Xcode

I know it's a bit confusing, but you didn't actually install Xcode 4. What you did was installing the Xcode 4 installer. So what you got to do is:

1. Open the Install Xcode-app located in your launchpad.
2. Make sure iTunes is closed, this is necessary, and the installer won't check for this.
3. Finalize the setup (well, Lion will do this for you, just press the buttons).
4. Optionally, restart your Mac.
5. Optionally, remove the Install Xcode-app.

It'll probably prompt you that it's moving your existing Developer-folder to Developer Old so it can install Xcode 4.1 to /Developer/.

It's a bit of a strange question to ask if you aren't experiencing any problems.

Yes, that's how you install Xcode when downloading from the developer centre. However unless you need to run a beta version you're probably better off installing Xcode from the App Store, where you can benefit from update notifications and delta upgrades.