New Lock, do I need a sensor too?

As far as which sensor, there are many different factors that go into this choice, including cost, whether you want zigbee or zwave or something else, personal aesthetics, etc. Also whether you wanna temperature sensor built-in.

The Device class features FAQ covers many of the different types:

Bulbs, switches and sensors, oh my....what to buy (device class features FAQ)

If you have the money, it’s hard to beat the Sensative strips. Zwave, available In both the US and the UK. These have a patent on a very Slim long lasting battery, so you can put them in the door frame and paint them and they will be nearly invisible. 10 year battery life. Works well with smartthings in either app. But expensive, usually around $60 each. Sometimes on sale for around $50.

Aeotec Makes a recessed sensor where you drill a long hole in the door frame. So this one is also invisible, but more work.

After that there are lots of different brands at lots of different prices. Smaller ones are usually more expensive.