I have a Verizon account and a Droid X. Used to be a great phone, and then Verizon updated it with their version of Android's Gingerbread O/S. It has been flakey ever since and a great advertisement for the iPhone.
If you go with any smart phone, go with the iPhone. There are many better Android phones (HTC comes to mind) but there is a long history intuitive design and reliability in the iPhone. However, both phones mean you will be paying Verizon for some sort of data plan, and you starting this thread asking for simple.
Simple, non-smartphones are quickly becoming a niche market. You are probably talk ins something from LG at the Cosmos. It has no 3G / 4G. It does a a weak browser that can cause data charges, but its slow enough you will not want to spend much time on it.
It does text, has a nicer camera than my Droid X if you want to snap pictures of your bike or hawt wife/girlfriend. But there are only a few low-end phones left. (The carriers all want the data plan dollars.)
I recommend going to a large, company owned Verizon store if you can find one nearby. Take a look at that they have The low end phones will be clustered together, and there will be more of them than at strip mall stores. If you do not have a company store nearby, log into http://www.verizonwireless.com
and go to the device upgrade page. They have filters to help you find non-smart phones.
For simple, go with the iPhone. For cheap and simple, go with the low end LG Cosmos or similar model. You might want to try a Droid, but Google is aiming to lock them down just as tight as the iPhone. They seem to want to nullify the advantages that the Droid market has. Rihgt now, It is next to impossible to root a non-rooted Gingerbread Android phone, and from the spec, the IceCreamSandwhich version of the O/S is going to be even tighter. If you want to be very afraid, spend some time browsing http://www.droidforums.net
and see why the interests are. I am the biggest geek I know, and some of those posts are beyond me.
There are now more Apps on the Android Market that the iPhone App store. There is very little control of what posts there, so not much prevents malware from creeping into a small percentage of them. There are few, if any Apple App store infections, but you pay for that by having to live with Apple examining every application, which takes longer, and excludes some for their own business reasons. Apple has a rigorous (some say ridiculous) control over the GUI and they live to make it simple in intuitive. Whether you like it or now.
iPhones take the best pictures. But I have the perpetually blurry Droid X.
The Android Phone market is unfortunately the clear leader in malicious software, which ranges from spam worms to viruses to bluetooth exploits. The iPhone has a few, and both browsers can get you into trouble, just like the one on your desktop can.