Haier electric dryer will light up, but when the start button is pushed you can hear the click in the relay box but the machine won't start.

Most likely it is dirty and lint deposits have built up, as it always does, in all the air ducts which will cause the main safety to blow causing the problem you described.You need a multi-meter to find which safety has blown. You will have to take it all apart and clean the drier out thoroughly or it is all for nothing. Every air duct will have to be removed and scraped and washed out to remove baked on lint deposits which bakes on like a concrete lining. This will burn so the manufacturer has installed the safeties to keep from having a house fire. The safeties are oval-shaped with a round center part containing the safety with two plug-on terminals for the wires and when they burn open, from the dryer being dirty, nothing will work or it will not heat. Using an ohm meter you put a lead on each terminal of the safety, with one wire unplugged, and if it is good you will have a reading of about zero; if it is bad it will read infinity. Driers were the number one cause of fires at one time, hence now we have safety fuses. Look at: How to clean your clothes dryer. How to make it hot again. - YouTube (http://www.youtube.com/watch?v=MMP7BW1lLs4) to see how. Also look at these to get an idea about how to clean it: Mr. Appliance Tips: How to Clean Your Dryer - YouTube (http://www.youtube.com/watch?v=iBiRw-KElC4) ; Cleaning a Dryer Vent and Lint Trap - YouTube (http://www.youtube.com/watch?v=yFTZGgVisCI)
Electric Clothes Dryers (http://www.electrical-forensics.com/Dryer/ElectricClothesDryers.html) Dryer vent cleaning prevents fires - YouTube (http://www.youtube.com/watch?v=A6wmD502-xE) and: Cleaning out a dryer with top lint cleanout - YouTube (http://www.youtube.com/watch?v=OBNDbyRm68M&feature=fvsr)
Cleaning out a dryer: By a woman 2 of 2 - YouTube (http://www.youtube.com/watch?v=a73HTFkcB90&feature=relmfu)
When you finish cleaning the drier it is best to also replace the belt, too, because you do not want to have to disassemble it all over again and if the belt is over two years old you should replace it.

Note that on most of the videos it does not show them washing out the parts, but this is important because it's the only way to really get the parts clean, just don't get motor wet or the capacitor if it has one. Vacuum out the motor to remove all dust. I would move the drier out to your garage to have plenty of space to work. You also want to check your vent and clean it, also. The vent should have a short run and be straight as possible. The job is really easier than it sounds. There is no shortcut that will work; it has to be cleaned well. Before you re-assemble it make sure to use white lithium grease (high-temp grease) and grease all bearings. Just vacuuming out the dryer a little does no good.

Is the main safety the same thing as the relay? I called Haier they don't have any part called the main safety?

The main safety is also called the high limit and is wired in series with other safeties like the door switch. You need to follow the wiring until you find it. It's not rocket science, but if you are trying to buy parts before you do your testing with a meter you are doing things backwards. Since there are often several safeties it is best to use a meter to find the right one before just randomly changing parts. If your car will not start you don't just buy any old part and put it in, do you? The same principle holds true for appliances. Use a meter and find out if all the safeties close. Check the door switch and make sure it works, check your start switch and make sure it closes when you press the button, then check the safety and centrifugal switch on the motor and move it manually to make sure it closes as it should. Here are some manuals that have wiring like your dryer: https://dl.dropboxusercontent.com/u/8010163/GE_Dryer.pdf
https://dl.dropboxusercontent.com/u/8010163/Whirlpool%20Dryers.zip
https://dl.dropboxusercontent.com/u/8010163/GE%20Dryer%20%20Manuals.zip
https://dl.dropboxusercontent.com/u/8010163/How%20to%20use%20Test%20Equipment%20Service%20Manu al.pdf
https://dl.dropboxusercontent.com/u/8010163/Amana_dryer.pdf
Pull the dryer out into a place that you can work on it like a garage. Take off the top and pull out the drum so you can see everything. Then with the meter set to "ohms" check every thermostat and safety by pulling one wire off each time you test something. When you find the one that reads "infinity" you will have found the bad part and then replace that part and only that part. Check all other parts, too. All the safeties and thermostats are wired in series so you just go from one to the other until you find it. Make sure and clean all of the ductwork including the blower and the housing. Take it apart and wash out the caked on lint deposits which is like concrete. Clean the part that holds the filter and if it has a door diverter, clean it too. . Lubricate all bearings with white lithium grease and get a new belt and install it. You will now have a like new dryer.