How do I identify get rid of virus/malware/something bad?

Question

How do I know what has gotten into my computer, and how do I get rid of it?  Got little weird pop-up messages that said if it showed up several more times, it meant my "PC" was infected with a virus.   I tried to close it out and it wouldn't let me.  I tried to Force Quit things, and at some point it threw a bigger window up on the screen, and right on top of that came a photo of a girl in a bikini asking me when I wanted to eff someone. 
Using Mavericks 10.9.5.   Have just installed and run a Norton anti-virus program, which said I have no viruses.  I realize that isn't the best software around, but it's free from Comcast and the first thing I could do.   I haven't had anything like this happen since I was on a PC
I was getting close to paying bills online, which clearly isn't the thing to do now.   What are my options as a low-tech user, and what are my official Apple options?  Past Apple Care coverage date.   Tried to call to find options through them, was told there was a long hold time, then got disconnected. 

Answer 1

If you saw a fake alert that "suspicious activity might have been detected," take Step B. Otherwise, take Step A when needed. In either case, remove the worthless Norton product according to the developer's instructions.
A
It's not malware. It's a JavaScript scam that only affects your web browser, and only temporarily. There are several ways to recover.
1. Some of those scam pages can be dismissed very easily. Press the key combination command-W to close the tab or window. A huge box will pop up. Press the return key and both the box and the page will close. If that doesn't happen, continue.
2. Press and hold command-W. You may hear repeating alert sounds. While holding the keys, click the OK button in the popup. A different popup may appear, which you can cancel out of as usual.
3. From the Safari menu bar, select
          Safari ▹ Preferences... ▹ Security
and uncheck the box marked Enable JavaScript. Leave the preferences dialog open.
Close the malicious window or tab.
Re-enable JavaScript and close the preferences dialog.
4. If the Preferences menu item is grayed out, quit Safari. Force quit if necessary. Relaunch it by holding down the shift key and clicking its icon in the Dock. None of the windows and tabs will reopen.
After closing the malicious page, from the menu bar, select
          Safari ▹ Preferences... ▹ Privacy ▹ Remove All Website Data
to get rid of any cookies or other data left by the server. Open your Downloads folder and delete anything you don't recognize.
B
You may have installed the "Downlite" or "VSearch" ad-injection malware. Follow the instructions on this Apple Support page to remove it.
Back up all data before making any changes.
Besides the files listed in the support article, you may also need to remove this item in the same way:
~/Library/Internet Plug-Ins/ConduitNPAPIPlugin.plugin
One of the steps in the article is to remove malicious Safari extensions. Do the equivalent in the Chrome and Firefox browsers, if you use either of those. If Safari crashes on launch, skip that step and come back to it after you've done everything else.
If you don't find any of the files or extensions listed, or if removing them doesn't stop the ad injection, then you may have one of the other kinds of adware covered by the support article. Follow the rest of the instructions in the article.
The problem may have started when you downloaded and ran an application called "MPlayerX." That's the name of a legitimate free movie player, but the name is also used fraudulently to distribute VSearch. If there is an item with that name in the Applications folder, delete it, and if you wish, replace it with the genuine article from mplayerx.org.
This malware is often found on illegal websites that traffic in pirated content such as movies. If you, or anyone else who uses the computer, visit such sites and follow prompts to install software, you can expect more of the same, and worse, to follow. Never install any software that you downloaded from a bittorrent, or that was downloaded by someone else from an unknown source.
In the Security & Privacy pane of System Preferences, select the General tab. The radio button marked Anywhere  should not be selected. If it is, click the lock icon to unlock the settings, then select one of the other buttons. After that, don't ignore a warning that you are about to run or install an application from an unknown developer.
Still in System Preferences, open the App Store or Software Update pane and check the box marked
          Install system data files and security updates
if it's not already checked.