The smart advice when it comes to troubleshooting heat pumps is leave it to the experts. There is not a whole lot of user serviceable parts in modern units so if you do not know what you are doing then you had better leave it alone.
That said, there are some things that you can try before calling out the engineer (or at least to give you some intelligent questions to ask him or her when they come out). Remember though that these units are coursing 240volts ac, and death by electrocution is possible.
If the heat pump unit simply does not start then you could look at the basics - check that none of the switches or the thermostat has tripped a fuse. There is some bypassing that can be done on a temporary basis to the non-vital parts - the time delay relay for example. Never bypass the low pressure or high pressure controls. If this paragraph does not make sense to you - do not touch anything.
Simple things you can do without damage are checking the refrigerant - too much or too little can cause the compressor to run constantly. Also check for anything obscuring the condenser - leaves and so on (if you disconnect from the mains you can clean it externally using a hose pipe).
If on the other hand the compressor runs really short cycles, it could be an installation problem - the unit could simply be too big for your home, or the thermostat could be badly placed. A couple more reasons to use an installer who knows what they are doing.