Why is it necessary to wear safety goggles and a lab apron while performing experiments in the lab?
In order to protect your mince pies.
Wearing safety spectacles, or regular optical spectacles, in a laboratory is an absolute necessity. Should you get a drop of hot nitric acid in your eye, that's your eye gone, and wearing spectacles continuously will prevent this.
The use of safety goggles is in my opinion a bit of overkill in a teaching laboratory. They are a bit bulky and cumbersome, and you are always inclined to take them off (I know I would be so inclined). Of course you have to follow the accepted procedure.
Laboratory coats are not really for personal protection. It is to protect your clothing from getting holed - look at the state of most lab coats. On the other hand, the use of aprons may be advizable if you deal with highly corrosive reagents.
Anyway, in whichever laboratory you work, you have to follow the procedures that are already laid down (and of course you also have to communicate these procedures to new members of the lab; i.e. instruct them in laboratory lore!). If you find them onerous or restrictive, then talk to your supervizor. But I reiterate that eye protection is an absolute necessity - most chemists I know that don't wear optical spectacles have developed a habit when they walk in a lab of donning the pair of safety specs that they do have, or taking a spare pair immediately from a spectacles tray. This is a very good procedure to adopt.
Eye protection is important in another respect. Cut yourself, or scald yourself, or even break an arm or leg, and basic first aid will help you. You might be surprised in the event of such an accident. There is usually someone (or several) clued up enuff to give effective assistance. Get an eye injury, and the first aider can DO NOTHING; the only thing he or she can do is get the injured person and stick them in the eye shower, and keep them there, and try to flush out the mince pie with copious water. In the event you should at least try to keep the unfortunate person irrigated for 15-20 minutes (I kid you not); at least until the paramedics get there.
Just to add that these days laboratories and schools and universities are much more safety conscious. They often offer gratis training in first aid and CPR (and of course this is useful outside the university; it's all too easy to choke on a piece of food that goes down the wrong hole). Try to take advantage of these offers; they look good on a CV, but even more importantly they might save someone's life.