Preferably a phone with email, WiFi and internet capabilities. Without signing up for a plan.

You really can't go wrong with any smartphone on a pre-paid plan, unless you go with a really cheap smatphone with low specs. If you are an Apple fan, then I would recommend getting an Iphone 5c, after Apple releases the Iphone 6 this month. If you are an Android person, then the Moto X from last year would be a good choice because of the release of the 5.2 inch verison of the Moto X, just a couple of days ago, as you should be able to get good pricing on the previous model Moto X (which has a 4,7 inch screen).

Walk to Wal-Mart (park off site to avoid video surveillance linking you to a vehicle) while wearing a hoodie and sunglasses, buy a pay-as-you-go SIM and pay cash with well-circulated, low denomination bills.

Wait 60 days and then go to a flea market in another town and buy a phone with the stipulation that it has to work with your SIM card first.  Pay cash after proving it works.  Bonus points if you take a borrowed car or catch a ride to the flea market.

Immediately disable Bluetooth and WiFi.  Make sure your fingerprints are not on the SIM or inside the phone; anyone can pick up a phone and leave a print but a print INSIDE is hard to deny.  Activate encryption on the device and use a very strong screen lock pass phrase.

Do not travel with the phone turned on: plenty of major roads & intersections in the US have cameras, and a cell phone signal trace can be used to put you in a location & time matched against a camera.  Do not turn on the phone in your house.  Do not turn on that phone if you have another phone active with you, or in a vehicle with an "On Star" service.

Do I do all this?  Fuck no, I'm not that paranoid and know my rights in a country that mostly respects those rights.  However, I've met people to whom these actions save their very lives from oppressive governments seeking to kill them for their ancestry, religion, or political views.  Spend time talking with someone who survived the Stasi in Germany, or the purge in Armenia, or modern day North Korea, Syria, or Egypt and you'll begin to understand.

Did you know that house phones in Scandinavia used to have a meter to track only usage?  They did this so there was no record of who/when/duration of your calls, only the time used and this was directly related to political persecution.  

Anyway, following the above nets you a phone & service that you can drop in a trash can without a second thought.  At worst you're out less than $200 and peace of mind is cheap.

Get a lower-end android at walmart on tmobile's prepaid plan. Pro tip: tmo has a walmart-only plan that doesn't include many minutes (100/mo) but has plenty of 4G data. $30/mo.

Pay cash for the phone, pay cash for the refill card.  

They may want an address or ssn to register the number when you first sign up, but just make something up.  

However, if I was going to have a drop phone for this hobby, I'd probably get a free or $10 tracfone.  

Or, if you're very techinically-minded, you can get legal, free Google Voice numbers that you can use in your browser.

I use this. You can pay cash to renew at a kiosk in their store  or buy top up cards nearly anywhere. I brought my own iPhone 4s that was unlocked and they put in SIM card and activated in 5 minutes. They asked for name, but don't check id.  

Get a compatible phone from sprint off the flee market or ebay and freedompop can be your hobby fone for free. It's a bit of a hassle to find a compatible phone that gets activated but it's possible to bring your own device.  

i prefer Gvoice and burner app tho. It's traceable since it's linked to google or google payments  
but so are others if you really think about it.  

I'm not extremely paranoid since I've been hobbying without a single incident using all types of hobbyphones. I change numbers regularly though.

so I just use a dumb trac phone.  I have had to change numbers twice.  I was on the verge of dropping the phone in a trash bin or under a truck tire.  Problems occur just often enough that it should be a consideration.

I use my existing unlocked smart phone with an AT&T Go Phone SIM card. I find it easier to deal with two SIM cards than two separate phones. I keep the extra SIM card in the battery compartment and switch cards whenever I get the itch.