I have an MyTouch 4G from TMobile. It's running android 2.3.4. How do I put put a file and delete files from the phone itself, not just the SD card?
You could install the Android SDK on your computer. This would give you the possibility to access your device's storage via the command line using e.g.
ADB FileExplorer is a minimalistic frontend to copy files from/to the device. It's freely available from the XDA developers (simply follow the link). It is written in Java, and thus should run on nearly all operating systems used on desktop computers.
If you want a little more comfort, then there's also ADBBrowser. Available for Linux and Windows (32-bit and 64-bit), it offers also an app-browser. And last but not least, there's QtADB available for Linux, Mac and Windows (again 32-bit and 64-bit), full of useful features like file manager, app manager, device information, SMS, shell, screenshots, logcat, backup/restore... Take a pick.
Of course, there are many more possibilities. If your device is rooted, there are several apps available on the Playstore which permit full access to the local file system remotely.
If you mean the Internal SD Card (where pictures etc are stored if they aren't stored on the internal SD Card), then this is mounted along with the external SD card when you connect your phone to the computer.
If you mean the root file system, (I.E where the Android system files are), then this can be done, but not directly.
You would have to do it using an app via FTP, and you would have to be rooted to see the vast majority of the files.
The app SSH Droid allows you to turn your device into an SSH/SFTP server, which allows you to use FTP to connect to the internal file system (where the main system files are stored) of your device, using an FTP client (Nautilus, Windows Explorer, Filezilla for example).
When doing this, be careful not to delete any files that could be important in the functioning of your device, and like I said - root will be needed to do most things here.