I have my first project running now, a temperature sensor using Arduino Uno, LM35 and a 16 x 2 LCD display.
I want to move the project from the Uno to a breadboarded ATMEGA8-16PU (to free up the Arduino board for other projects).
I can manage a 5V power supply for the breadboard, but I'm unclear on how to program the Atmega8. Can I just stick it on the Uno board in place of the Atmega328 and program it, then transfer it to the breadboard? Or do I have figure out how to program the Atmega8 on the breadboard?
"Can I just stick it on the Uno board in place of the Atmega328 and program it, then transfer it to the breadboard? "
Yes, select "Arduino NG or older with Atmega8" as the board type.
Look in the Hardware section, Index of older boards, to see what the Atmega8 used.
If you have an AVR ISP you can program the bootloader, then use normal Serial programming for downloading sketches.
You can also use the Uno to bootload the part on a breadboard, you will need whatever external components the "Arduino NG or older with Atmega8" uses.
I'd recommend you get a real AVR ISP if you're going to be doing more projects. Having a few tools make life easier.