The easy bit first:
'Dragging & dropping' won't delete the source files from their original locations.
Next, I'll divide the stuff on your hard drive into three categories:
1. Programs:
You can't just copy a program from one computer to another. Each program has to be installed separately onto each computer. So there's no point transferring program files. However if (say) you've downloaded a set-up file for a particular program onto your hard drive, then you can copy that file across and run it on your new computer to install the program.
2. Data which is accessible by any suitable program:
Pictures on your computer are available for use by any image-handling programs, such as Windows Photo Viewer, Picasa, Irfanview, Photoshop or whatever. Because they're not 'within' any particular program, they can simply be dragged-&-dropped without any problems. The same applies to documents, spreadsheets, databases, etc.
3. Data which is associated with a particular program:
Your browser bookmarks are only available for use within that browser and can't simply be opened by any other program. To transfer them from one computer to another you need to 'export' them to the external drive and then 'import' them back from it. In Firefox, for example, you need to go to Bookmarks > Show All Bookmarks > Import & Backup to find the relevant options.
Similarly (and importantly) emails held by programs such as Outlook Express or Thunderbird aren't accessible by other programs. Once again, you need to locate the options to export, and then import, the data. (Clearly that won't be relevant if you're using a web-based email service because your emails are stored on a remote server, not on your own computer).
If you need specific instruction for any particular program there will be people here who can help but the information should also be available via 'Help' on a program's menu bar (or by pressing F1)