Use Dallas DS1821, digital 1-wire thermostat temperature sensor along with PIC or any other microcontroller. DS1821 is a good sensor with +-0.5 degree celcius resolution.
Here you can find some source and schematic of DS1821 https://www.microchipc.com/sourcecode/#ds1821
A bit of both, really. Often I take snippets of schematics, and put them together (like Java Beans, if you're into programming). There are also a lot of circuit encyclopaedias available at most libraries, and quite a few good circuit sites on the web.
As for the LED readout for your thermometer - have a look at the LM3914 - it seems to be popular lately It's an LED bargraph driver.