Avoid storing propane tanks indoors and especially don't store them in an area that is air conditioned and then pull them outside and hook them up to the grill in 90 degree heat (and vice-versa). The extreme fluctuation in temperature could cause them to explode.
Check for leaks - mix dish detergent and water in a spray bottle and spray the fittings and the regulator when turned on. If you see any soapy bubbles form your have a leak and need to disconnect the tank, close the valve and take the tank to the proper authorities to be dealt with. DO NOT use a tank that has a leak.
Read and understand the owners manual for your grill. Know it's safety features and follow the instructions for lighting it to a tee. If the grill does not light TURN EVERYTHING off and walk away for a few minutes to let any propane in the area dissipate.
Make sure the grill is on a level surface and has clearance all around it. A hot grill can warp the siding on your house even at several feet so pay attention to the area you place your grill in.
NEVER, and I will repeat NEVER, use a grill indoors. They are not made for use indoors and you can die doing this.
Keep your fire under control, avoid excessive flare ups.
Wear tight fitting clothing when grilling, this also goes for when cooking indoors. A loose sleeve or other piece of clothing could ignite if you are not careful.
This may seem obvious but grills get HOT and they stay that way for a LONG time after they are turned off. Let your grill cool before moving it, cleaning the outside of it, or COVERING IT. I have seen a neighbor have a cover melt to the grill because he was impatient. DON'T COVER IT UNTIL IT IS COOL TO THE TOUCH!
Grilling is a great way to have fun and cook great food, grill safely and you will ensure you and your family enjoy the summer, worry free!
...that is all.