I'd like to second the other posts - if you are familiar with Arduino, going to a stand-alone chip is very easy.
You miss out on the following things: programming via USB, 7-12volts through the power jack, and/or power through USB, reset button, led on pin 13, and the clock. Depending on where you get your Atmel chip, you may also need to set the fuses.
The code is exactly the same, even if you go down to a smaller chip (or larger - whatever suits your project best!).
Programming: I use a FreeTronics ICSP (ISP) to program the chips, and have the 6-pin ICSP header on all my projects for easy reprogramming. This particular programmer has a "slow clock" jumper, because when you get the chip from the Atmel factory, their internal clock runs at 1mhz, too slow to program with the Arduino IDE default settings. You can leave it at 1mhz, if your project doesn't need any more, or change the fuses. You will need an ICSP to set the fuses (see further down). By using an ISP rather than usb, you also save a little memory - you don't need a bootloader (which is pre-programmed on Arduinos, but not when you buy the chips from the factory).
You will need to sort out your own power - this depends on what you need, and what you have. You can use a 5-volt AC-DC power supply ("Wall wart") and plug it into mains. Or, you can power it off USB - if you only use USB for power, you can solder the socket directly on your board. You can use a regulator (which is what the Arduino uses for the 7-12v power socket), or, if running it off a 12v power supply, it might be easier/cheaper to buy a car phone charger from the dollar store, and a lot of the work will be done for you. Also, the power supply doesn't have to be 5v - depending on your application, you might run it from 5.5v to as little as 1.8v, depending on your clock speed. Find the data sheet for the chip you use.
A reset button is very easy to add, if you need it - all it takes is a resistor and a pushbutton. Weather you need it or not, depends on your project.
The same applies to the led on pin 13 - it depends on your project.
Finally, the clock. The Arduino has a 16mhz quartz clock. You can do the same, it only takes 3 components (you will have to set the fuses, see further down). You can use a clock up to 20mhz, giving you a 25% speed boost if you need it (you may need to change some settings, because timing functions like delay depend on the clock speed - your delays will be 25% shorter). Or, you can use the internal 8mhz clock (either at full speed, or as from factory, at 1/8 speed, so you get 1mhz). At 4mhz or less, you can run the Atmega at 1.8 volt, or 0.7 volt for the "pico power" chips.
As I have mentioned, you may need to set the fuses to the correct value. The best idea is to go to http://www.engbedded.com/fusecalc/ to work out what fuse settings you want. Specific things to look for: the clock source (internal or external? what speed?), divide clock by 8 internally, and brown-out detection (Arduino chips turn themselves off when the power goes below 4.3v). Warning: setting wrong fuses may set your chip so it is very difficult to fix. You can set the fuses using an ISP; if the clock speed is very slow (I think < 4mhz? which it is when it comes from the factory), then you will have to "slow program" - some ISPs have a "slow program" jumper, but you can use an option to avrdude to do the same - I think it's -B e.g. -B10).