When you need to switch a lot of power, N channel MOSFETs are best for the job. These FETs can switch over 60A and 30V and are TO-220 package so they fit nicely into any breadboard or perfboard. Heat sinking is easy with TO-220's, but because of the very low Rds(on) of down to 0.009 ohms (depending on the Vgs - check the datasheet) you can get away with no heatsinking for pretty high loads! The threshhold voltage is also very low, less than 2.5V so you can control it directly from a microcontroller running on 2.8V, 3.3V or 5V logic.
The TO-220 package can dissipate up to 2 Watts without a heat sink (at room temperature). Because it has an Rds of as low as 9 milliohms, that means that you can switch at least 15 Amperes without a heat sink - more if you use PWM. For more detailed calculation, we suggest this site which has always worked nicely!
Driving LED strips with Arduino PWM with power FETs