One of the nice things about microcontrollers is that you can pretty much wire buttons directly to them. The AVRs upon which the Arduino is based have integrated pull-up resistors, so you can literally just wire a button between an IO pin and ground, turn on the pull-up resistor, and read the input with a 0 being "button pushed" and a 1 being "button not pushed". I'm guessing the Arduino libraries further automate this for you (I generally just program the "bare metal" AVRs with none of the Arduino libs, so I'm not sure of them). So yeah, a start button is no problem.
For transmissive/interruptive IR solutions, you've got a few options. You can buy discrete IR LEDs and phototransistors. These aren't too hard to work with. Basically, the phototransistor will act like a closed switch when it sees IR light in abundance and will "go open" when the IR light is blocked. The light source would then be an IR LED that you wire up just like a normal LED except you can't tell when it's on because you can't see it

You can also get somewhat more "integrated" solutions that add some analog signal conditioning to the detector side to give a true "logic level" output. Might make your life a little easier, but those tend to be like 20 bucks a piece for some strange reason. Sparkfun has a number of IR options on
their IR sensors page. You can also start poking around at good ol'
Digi-Key,
Mouser Electronics,
Allied Electronics and other places.
There also exist reflective IR sensors. They're mostly used when you want to sense something without having access to put a receiver on the other side (but maybe you can put a reflector on the object you want to sense or a passive reflector on the other side). Probably not necessary in this case, and those are more complicated to design for.
For hooking up to your Arduino, you just want a digital on/off or "logic level" output from the sensor. A bare phototransistor with a weak pull-up resistor on the collector like the internal one to the AVR or about a 10k external one and grounded emitter will probably give you something pretty close to this at low cost.