How do you avoid internet romance scams?

Don't send money to somebody you only know from the internet, or phone calls, period.
There is no way around this. The person who is asking you for money will not die, or come to harm, or you will not miss out on millions of dollars they are promising to give you. Don't send money!
Always expect to see a face on webcam, and learn to use Google Image search!
How stupid are you going to feel when you finally realize the person you are in contact with is fake, and using stolen pictures? Some people will never accept the truth, and will die thinking the famous celebrity in the pictures used by the scammer, who actually is a millionaire, stole a few thousand dollars from them. Don't be gullible. Expect to see a face on cam. This will filter out most scams. (although some scammers do have video of 'that face' they can play over a webcam, making it appear that person is live.)

RESEARCH! Along with getting in the habit of checking pictures you see on dating sites with Google Image search, do a search of email addresses of new 'friends' you meet on dating sites. You may be surprised what you find.
Do you really want to feel stupid? Fall for a scam that has been reported online previously. Same face, same story. These scams are not sophisticated. They are fairly simple. They count on the victim being trusting, and gullible. Don't be that victim!

I also use SPOKEO. It lets me see any possible social network sites that email address may be connected to, pictures, and more.
Search anybody by name, e-mail address, phone number, online username or even friends in your address book and instantly return lots of info.

Why is it important to run a google search of the email address of a new 'friend' you meet online?

Scammers change email addresses like we change underwear, but doing a search of an email address for a new person you meet online may help you avoid getting scammed.
We try to teach people to post information about scams, and scammers, online. This means many email addresses used by scammers will show up in the search engines.
A scammer generally doesn't easily abandon an email address he or she is currently using for contact with victims, or potential victims. When they do, they will usually claim the email account was 'hacked', or they otherwise lost access to it.
Please, do a google search of the email address for the new person you are in contact with. You may find nothing, you may find many things that don't add up to the story they tell you, or worse.
Thousands of 'baiters' and other anti-scam activists put the information about scammers online for this purpose, to help you, if you bother to look.
You can additionally use other search services, like Spokeo. Spokeo will tell you a lot about real American people you meet on the 'net. Search anybody by name, e-mail address, phone number, online username or even friends in your address book and instantly return lots of info.