I'd recommend a cheap laptop. You can get a 15" laptop running Windows 8.1 for under $300, and Office 365 Personal for $70/yr or Office 365 University for $80 for 4 years (if you qualify as a student). You will need to reinstall your programs and games on the new computer, and you will probably need to reconfigure your settings for all your programs. As far as transferring documents and files, just put everything on a USB stick to copy it over.
If you decide to go the desktop route, you're typically looking at more expensive, but more powerful systems, with monitors not generally included, and without touchscreen. If you're looking for office productivity or higher-end gaming, you'll be looking towards desktops; otherwise, you should stick to laptops.
If you're buying a monitor, Dell is a good brand but just about any brand is equal in terms of reliability. As standard features, monitors will be LCD widescreen supporting 1080p (1920x1080) native resolution. Monitors start at about $100+, and you can usually get up to 24" in the $100-$200 range. Any monitor you buy should support both DVI and VGA connections.
In my experience, it always helps to price-compare online between FS, BB, Staples, and Dell before making a purchase, as all of them always have sales going on.