It's a Frequently Asked Question :)
Removal of your private data is priority. FAQtually can erase your old computer's hard disk drive (HDD) to U.S. Department of Defense standards, ensuring none of your data can be retrieved by anyone. Removing the HDD and storing it in an antistatic bag as a data backup/archive is another option. Also, you can remove the HDD, disassemble it and take a hammer to the disks, physically rendering them nonfunctional. Finally, you can restore* the computer to its new-from-the-box state by using the recovery CDs, DVDs or HDD partition.
Once your data has been removed from your old computer, you can safely donate it, sell it or dispose of it at your local consumer hazardous waste disposal repository.
*Restoring doesn't necessarily destroy the data, but it does make it reasonably unreadable for giving the computer to a friend. Were I to donate or sell a computer, I would completely erase the HDD before restoring it (works only with restoring from CDs & DVDs) or I would donate/sell it without the HDD.
Yes. (Let's qualify that.)
Windows Updates (those that actually update your Windows XP, Vista, 7 operating system) are very important. Yes, they occasionally (very rarely) cause problems with very old or proprietary programs, but can usually be undone if problems arise. The benefits of updating Windows far outweigh the risks. Not updating Windows leaves potential security vulnerabilities unpatched and can result in major headaches due to malicious programs or activities.
Adobe updates are probably popping up notices on your system more than anything else. Apply the updates. Yes, the frequency of them is irritating, but they are usually addressing security vulnerabilities in Adobe Reader that could cause problems. Yes, Adobe is going to put its Reader icon back on your desktop every time it updates. It's OK to delete that icon every time, too. Have you ever opened Adobe Reader first and then opened the PDF you want to view? I haven't! :)
Java is probably the next most frequent update notice seen on Windows PCs. Update it. Be sure to pay attention to the prompts when updating; they often try to add something extra (like a toolbar for Internet Explorer). It's OK (and recommended) to uncheck those options.
Printer manufacturers (HP, in particular) often have update notices. These are almost always safe to skip, though updating may yield new functionality. Read any details about the update if you are unsure.
Apple has an autoupdate function, too. Most often, it's offering updates to iTunes and Quicktime. It's OK to update these, though skipping them is often fine, too. Updating iTunes often provides new features. Apple's updater will also try to get you to install Safari on your Windows PC. (It's already on Macs.) You do not need this. It's OK to have it, but I bet you won't use it. :)
YES! Unless the files on your computer are easily replaceable, you should back them up.
All those photos & videos of family and friends... All your music... All your schoolwork, reports, taxes, etc.... These can be easily lost forever. A hard drive failure, a virus, a fire, a natural disaster - all kinds of things can happen that can destroy your data. Sometimes the data can be retrieved, but the cost of data recovery can get very high and still not guarantee any or all of it is recovered.
Backing up to an external hard drive is strongly recommended. Windows and Mac OS have built-in backup programs and many external hard drives also come with backup programs. These programs are easy to use and often back up automatically for you as long as the drive is connected to your computer.
Offsite backups are your next line of defense against data loss. If your backups are kept where your computer is, what happens if there's a fire or other disaster? Having multiple external hard drives and rotating their use, storing only one onsite and the rest offsite (with family, friend, safe deposit box, etc.) is a good plan. However, it completely is dependent upon YOU staying on top of it. If you want to "set it and forget it," try Carbonite. It's super easy and you can't beat the cost. I had a hard drive fail a few years ago (right in the middle of developing a large e-learning project) and was so happy I'd installed Carbonite. Once I got the PC back up and running, I re-installed Carbonite on it and immediately downloaded the files I needed to continue my work. In the background, Carbonite put every single one of my other files right back where they originally were. It rocks!
NO! You do need antivirus, but you do not need to buy it. Microsoft Security Essentials is a top-rated FREE antivirus that's light on consuming your PC's resources. Other antivirus applications (often pre-installed on new PCs) are repeatedly costly, resource-hungry and can nag to you madness. MSE is a great solution for the majority of consumers and many small businesses, too.