I recommend you stay away from "cheap" hand held radios if you're meaning the sub-$50 Chinese variety. Yes, they are cheap but they are so inferior to a Yaesu, Icom, Kenwood, or Alinco in every way,. I wonder how many people decided to get into ham radio with a Chinese hand held and were so discouraged by the pour quality and horrible user experience that they abandoned the hobby. In my experience you need more power and antenna gain for APRS then for voice communication thru repeaters. I have much better luck getting my beacons received from a mobile rig then a hand held. Really, a hand held is a poor choice for a first ham radio. They have poor range, small buttons, small displays, etc. They seem like a perfect way to have a radio in your home, car, and portable all in one. In reality their range is extremely limited in a car so you add an external antenna. Then the battery dies from running it on high power so you get a 12 volt adapter. Then the radio keeps getting hot in your hand so you get a speaker mic. Now you have an octopus of wires in your vehicle and you still have a small display, small buttons, and only 5 watts. A mobile rig is better and often less expensive.
There are several Kenwood radios that have APRS built in. I'm more familiar with Yaesu; their FT1XD, FT2DR, FT3DR, and FT5DR all have APRS built in. So do several mobile rigs: FTM-100, FTM-300, and FTM-400. I personally have the FT1XD and FTM-400DR. The '400 is nice because it has a touch screen with a full keyboard so it's much easier to send messages over APRS. When I am not in cell phone range I can use my FTM-400 to send messages to my wife's phone via the APRS SMS gateway. She can then reply back from her phone. This usually works but there are apparently some rather misinformed but well meaning hams who have installed "receive only" digipeaters. When my APRS path from my radio to the SMS gateway goes thru a receive only digi then my wife's reply doesn't make it back to me.
You can also use APRS to send short email thru the APRS email gateway. I have done it to try it but I don't use it very often.
I also have the Mobillinkd TNC and a cable to connect it to my Icom IC2720 or Yaesu FT-7800. Both of these radios are just "regular" dual band radios but they have a DIN plug on the back to plug in the TNC. With the TNC plugged in I can connect to it with Bluetooth from my phone. This works really well too.
If you really need extra range you can do APRS on HF as well, usually 30 meters. And, you can digipeat APRS thru some amateur radio satellites too.