Tyres: Having suffered a blow out on landing I would express concern about "bald" tyres. The reality is the material underneath the tread is VERY strong so unless the runway is wet the tyres would have been safe. Like a cars tyres the "tread" helps prevent aquaplaning on a wet runway.
Fuel Gugae: ON a PA-28 they are about as much use as a chocolate teapot! I believe the regulations are for fuel guages only to be accurate when the tank is empty. As previous poster has said - ALWAYS look into the tanks. If you are low hours you may find landings easier with full tanks....
Personally I always make sure the tanks are topped up "to the tabs" every time I fly - even for a 1 hour local hop. Also remember you have to do fuel management on a PA-28 - the fuel tank switch is by your feet, and I am sure your instructor will tell you the correct way to switch tanks and "balance" the fuel you use.
Good luck - enjoy your training and don't ever be afraid to ask your instructor questions. When you are training the only dumb question is the one you don't ask. If you are uncomfortable with this try another instructor.
